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SSC CGL Tier 1 and Tier 2 Program

------------------------------------------------------------------------------------------------------------------- Section : Math Chapter : Geometry Days 59- 64

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Geometry

Significance:

7-8 questions can come in tier 1 and 15-16 questions in tier 2. This chapter will come under advance math What is advance math? Many people consider these four (geometry, algebra, Mensuration,

trigonometry) chapters difficult, many people consider high weightage in overall math. According to me nothing is advance just concentration and practice make you scoring these four topics. So I will call it as scoring math.

Atleast 4-5 questions will be directly related to properties. So understanding properties will be highly useful. Remember properties and should be on your tips.

What will we cover in Geometry chapter?

Lines

Angles

Triangle

Quadrilateral

Polygons

Circles

Let us start from lines

Parallel lines: those two or more lines which will never meet just like railway track lines.

Transverse line: A line which will cut parallel lines. It is shown diagrammatically below.

From a point N number of lines can be drawn. As many as you wish.

Understand angles making with lines.

AB and CD are parallel lines. Symbol of parallel lines is . You will find many places symbol of line like ̅̅ ̅̅ or simply AB;

both are same thing.

EF: - Transversal lines

Corresponding Angles

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∠1 = ∠5

Alternate angles

∠1 = ∠7 or ∠4 = ∠6 or ∠3 = ∠5 etc

E

1 2

A B

4 3

5 6

C D

8 7

F

Adjacent angles

∠1 + ∠2 = in this case as these are linear pair. Any two angles which are adjacent are called adjacent angles. It is

not necessary that their sum should be .

Sum of interior Angles same side = 2 right angle = 1800

∠3 + ∠6 = 180

Sum of Exterior Angles same side

∠1 = ∠8 = 180

Vertically opposite angles = ∠1 = ∠3 or ∠2 = ∠4

Internal Angle Bisector = angle divided in two equal parts

Internal Q

Q



External Angle bisector

Q Q

Angle = 8 Types

1. Acute An gle = 0 - 900

2. Right Angle = 900

3. Obtuse Angle = 900 - 1800

4. Straight Angle = 1800

5. Reflex Angle = 1800 - 3600

6. Complete Angle = 3600

7. Complementary Angle = Sum of 2 Angles = 900

8. Supplementary Angle l= Sum of two Angle = 1800

TRIANGLE

What is vertex (singular, vertices (plural) : Vertices A, B, C shown in diagram

How to show angles: Angle = ∠A, ∠B, ∠C

How to show sides: Three sides = AB, BC, AC

Triangle can be classified in two ways: based on sides and based on angles

Triangles are of 3 Types based on sides

Equilateral Triangle = all three sides equal

Isosceles Triangle = any two sides equal

Scalene = all three sides are different in length



Triangles are of 3 Types based on angles also

Right angle triangle : one angle will be of 90 degree

Obtuse angle : one angle will be more than 90 degree

Acute Angle : all three angles should be less than 90 degree

Congruency of Triangle ( congruent triangles are those that have similar size and shape ; this means

corresponding sides are equal and corresponding angles are equal)

Rules that help to decide congruency:

Two triangles will be congruent if :

SSS: (Side- Side – Side rule): When all three sides are equal

SAS: (Side – Angle – Side Rule): When two sides and one angle are equal



ASA: (Angle – Side – Angle Rule): When two angles and one side are equal

Hypotenuse Leg Rule / Right Angle- Hypotenuse- Side ( RHS) rule

Similarity of triangles

A P

B C Q R

; Perimeter of

=

A

P Q

B C



PQ ‖ BC ; P and Q are mid points of AB and AC respectively.

PQ =

In case of Angle bisector

P

Q S R

PS is common side ;

If DE intersect lines AB and AC at D and E then

A

2.5 2

D E

5

B 9 C

e.g. AD =2.5 ; DB =5 ; AE =2; BC = 9 ; then what will be DE and EC.

Value of EC will be 4 and value of DE will be =3

Altitude

You can call it height also



Orthocenter

A

F e

O

B D C

All three Altitudes meet at O

Angle Bisector

I = incentre

Circle inscribed in the triangle

A

F i E

B D C

ID = IE = IF ( will be radius)

∠BIC = 90 + ∠

for Internal Angle Bisector

A

B C

P



Bisector External Angle

∠BPC = – ∠

Perpendicular bisector

Mid points of side

Perpendicular to it

Circle circumscribed about the triangle

P

F E

G

Q D R

PG = RG = QG also these will be radium of the circle

∠QGR = 2∠P always angle formed at centre is double of angle formed at circumference

Median

Median : the line drawn from vertex to midpoint of opposite side



Each median is divided into 1 : 2 at centroid. ; Point of concurrency is called: Meeting Point ; G = Centroid

Apollonius theorem : It is related to Median

A

F G E

B D C

AB2 + AC2 = 2 (AD2 + BD2)



Other key points for median

A

3

D F E

1 1

2 G

2

B M C

Ratio of AF : FG is always in 3: 1

2

Area of

If area of 3 3 2 6

Other facts about triangle

C

A B

Sum of two sides is always greater than third side : e.g. AB + BC > CA ; AB + AC > BC ; BC + CA > AB

Difference of two sides is always lesser than third side : AB – BC < CA ; AB – AC < BC ; BC – CA < AB

Quadrilateral

Parallelogram: Opposite sides equal and parallel

D C

O

A B



AB = DC, AD = BC

∠A + ∠B = ∠B + ∠C = ∠C + ∠D = ∠D + ∠A = 1800

∠A = ∠C and ∠B = ∠D

AC , AO = OC, OB = OD

Rhombus

All sides are equal, parallel

D C

O

A B

∠A + ∠B = ∠B + ∠C = ∠C + ∠D = ∠D + ∠A = 1800

∠A = ∠C, ∠B = ∠D

4 =

Sum of square of sides = Sum of square of diagonals

Diagonals bisect Each other at right angle

Form four right angled triangles

Diagonals : Not of equal magnitudes

Four right triangles

Each equal

th of area of rhombus.

Rectangle

D C

O

A B



AB = CD

AD = BC

∠A = ∠B = ∠C = ∠D = 900

Diagonals are equal

Bisect each other

AC = BD

OA = OB = OC = OD

Square

Diagonals bisect each other at right angle Triangle

4 isosceles right angle Triangles

AB = BC = CD = AD

∠A = ∠B = ∠C = ∠D 0

Diagonals of square AC = BD

Trapezium

AB ‖DC

D C

A B

Only one pair of opposite sides are parallel



Cyclic quadrilateral

2

A 2

D C

Polygons

Convex polygon:- none of its interior angle more than 1800

Concave polygon: At least one angle is more than 1800

Regular polygon = Sides & Angles equal

Exterior Angle =

;

Interior Angle = 1800 _ Exterior Angle

In case of convex polygon, sum of all interior angles = (2n – 4)

Exterior Angles sum = 36

No of diagonal =

CIRCLE

Angles in same segment are equal



Angle in semi –circle: Right Angle

Two chords intersect each other internally

Or Externally AE

O

A

E D

C B

Internally

A

B

O E

D

C Externally



Tangents to a Circle

(i)

O

P T

T = Point of contact

OT ⊥ PT

(ii) A

P O

B

PA = PB ; P = outside point

From any outside point P, distance of both tangents is equal.

(iii)

A B

Q

Q

P T



Angle made by chord with tangent is always equal to the angle on any point of circumference.

∠PTA = ∠ABT

AT = Chord

PT Tangent to circle

(iv) Below is the important properties , frequently asked in SSC /competitive exams .

A

B D C

E

AB. AC + DE. AE = AE2

(v)

A

B

D P

C

PA



(vi)

T

P

A

B

PT2 = PB

P = External Point

T = Point of Contact

PAB = secant to the circle

PT = Tangent

PAIR OF CIRCLES

(I)

A C B

A, B,C Collinear

C Point of Contact of the two circles



(ii)

A

E

F

O d B

H D

L

C G

Direct Tangents:

Length of the direct common tangent (L) = √ ; d = distance between two centers, r and

(iii)

O d

d

Length of transverse Tangent (L) = √

(iv). When two circles touch then d = r1 + r2

Cyclic Quadrilateral

∠A + ∠C = ∠B + ∠D = 1800



Sum of opposite Angles = 1800

A B

D C

Project 400 Questions

Q 1.ABCD is a rhombus in which side AB = 4 cm and ∠ABC = 1200, then the length of the diagonal BD is (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm Solution : (d)

∠ABC = 1200 ;∠ABD =

6 0 ; ∠ABD = ∠ADB = 600 ; AB = AD ; ABD is an equilateral ; BD = 4 cm

Q 2. If I be the incentre of ABC and ∠B = 70 and ∠C = 50 , then the magnitude of ∠BIC is (a) 130 (b) 60 (c) 120 (d) 105 Solution :(c)

∠IBC

35 ∠ICB

25 ∠ 35 25 6 2

Q 3. In ABC, D is the midpoint of BC. Length AD is 27 cm. N is a point in AD such that the length of DN is 12 cm. The distance of N from the centroid of ABC is equal to (a) 3 cm (b) 6 cm (c) 9 cm (d) 15 cm Solution :(a)



AD = 27 cm; Centroid = O;

; =

2 ; ND = 12 cm; 2 3

Q 4. A tree of hight ‘h’ metres is broken by a storm in such a way that its top touches the ground at a distance of ‘x’ metres from its root. Find the height at which the tree is broken. (Here h>x (d) 20 )

(a)

metre (b)

metre (c)

metre (d)

metre

Solution :(b)

AB = Height of tree = h metre ; AC = Required height = y metre; BC = CD = Broken part of tree =(h – y) metre AC2 + AD2 = CD2 ⇒ y2 + x2 = (h – y)2 ⇒ y2 + x2 = h2 + y2 – 2hy ⇒ x2 = h2 – 2hy

⇒ 2hy h2 – x2 ⇒ y

Q 5. For a triangle ABC, D, E, F are the mid-points of its sides. If ABC = 24 sq. units then DEF is (a) 4 sq. units (b) 6 sq. units (c) 8 sq. units (d) 12 sq. units Solution :(b)

4

4 24 6

Q 6. The in-radius of an equilateral triangle is of length 3 cm. The length of each of its medians is

(a) 12 cm (b)

cm (c) 4 cm (d) 9 cm

Solution :(d)



In equilateral triangle centroid, incentre, orthocentre coincide at the same point. ;

;

3 3 Q 7. The radius of two concentric circles are 17 cm and 10 cm. A straight line ABCD intersects the larger circle at the point A and D and intersects the smaller circle at the points B and C. If BC = 12 cm, then the length of AD (in cm) is: (a) 20 (b) 24 (c) 30 (d) 34 Solution :(a)

BE = EC = 6 cm, OB = 10 cm, OA = 17 cm; From ; OE = √ √ 6 √ 6 4 ;

From ; AE = √ √ √25 5 ; 2 2 5 3 Q 8. The radius of the circum circle of a right angled triangle is 15 cm and the radius of its inscribed circle is 6 cm. Find the sides of the triangle. (a) 30, 40, 41 (b) 18, 24, 30 (c) 30, 24, 25 (d) 24, 36, 20 Solution :(b) Simply, check through options & find Pythagorean Triplet

182 + 242 = 302; 24 3 Q 9. Two circles are of radii 7 cm and 2 cm their centres being 13cm apart. Then the length of direct common tangent to the circles between the points of contact is (a) 12 cm (b) 15 cm (c) 10 cm (d) 5 cm Solution :(a)



Direct common tangent; √

; = √ 3 2

√ 6 25 √ 44 2 Q 10. If the radii of two circles be 6 cm and 3 cm and the length of the transverse common tangent be 8 cm, then the distance between the two centres is

(a) √ (b) √ (c)√ (d) √ Solution :(a)

Length of transverse tangent;=√ ⇒ √ ⇒ 64 XY2 – 81

⇒ XY2 64 45 ⇒ XY √ 45 Q 11. Two parallel chords are drawn in a circle of diameter 30 cm. The length of one chord is 24 cm and the distance between the two chords is 21 cm. The length of the other chord is (a) 10 cm (b) 18 cm (c) 12 cm (d) 16 cm Solution :(b)

AB 24 cm ⇒ AE EB 2 cm OE √ √ 5 2 √225 44 √ 2 2 ; Also, CF = 2 Q 12. Two equal circles of radius 4 cm intersect each other such that each passes through the centre of the other. The length of the common chord is:

(a) 2√ cm (b) 4√ cm (c) 2√ cm (d) 8 cm Solution :(b)



OC = 2 cm; OA = 4 cm; √4 2 √ 6 4 √ 2 2√3; 4√3 Q 13. Two circles touch each other internally. If their radii are 2 cm & 3 cm. Find the length of that chord of larger circle which touches the smaller one.

(a) 2√ cm (b) 4√ cm (c) 3√ cm (d) None of these Solution :(b)

PQ = 3 cm, OP = 2 cm OQ = (3 – 2 cm cm OC OQ QC ⇒ 2 QC ⇒ OC cm AQ 3 cm

AC = √ √ 3 √ √ 2√2 ; AB = 2AC = 2 2√2 4√2

Q 14. ABCD is a trapezium whose side parallel to . Diagonal and intersect at O. If = 3, = x –

3, = 3x – 19 and = x – 5, the value (s) of x will be: (a) 7, 6 (b) 12, 6 (c) 7, 10 (d) 8, 9 Solution :(d)

Clearly, ;

⇒

⇒ 5 x2 – x 5 ⇒ x2 – x 2 ⇒ x2 – 8x – 9x

2 ⇒ x x – 8)-9(x – ⇒ x – 8)(x – ⇒ x or Q 15. The length of the diagonal BD of the parallelogram ABCD is 18 cm. If P and Q are the centroid of the ABC and ADC respectively the length of the line segment PQ is (a) 4 cm (b) 6 cm (c) 9 cm (d) 12 cm Solution :(b)



Centroid is the point where medians intersect. Diagonals of parallelogram bisect each other.

OP =

3 ; OQ =

3 ; 6

Q 16. The sum of the interior angles of a polygon is 1440 . The number of sides of the polygon is (a) 6 (b) 9 (c) 10 (d) 12 Solution :(c) Sum of interior angles of regular polygon = (2n – 4)

2 4 44 ⇒ 2 4 44

6 ⇒ 2 4 6 ⇒ 2 6 4 2 ⇒

2

2

Q 17. An interior angle of a regular polygon is 5 times its exterior angle. Then the number of sides of the polygon is (a) 14 (b) 16 (c) 12 (d) 18

Solution :(c) If the number of sides of regular polygon be n, then each interior angle =

; And each

exterior angle =

;

⇒ 2 4 5 4 ⇒ 2 4 2 ⇒ 2 2 4 24 ⇒ n

2

Q 18. Ratio of the number of sides of two regular polygons is 5 : 6 and the ratio of their each interior angle is 24 : 25. Then the number of sides of these two polygons is (a) 20, 24 (b) 15, 18 (c) 10, 12 (d) 5, 6

Solution :(c) Let the number of sides be 5x and 6x respectively. ; Then,

;

[

] ⇒

⇒

⇒ 25x – 10 = 24x – ⇒ x

– 8 = 2; 2 Q 19. ABCD and ADEF are parallelograms in the given figure. If CA = AF and ∠ACD = 600 then DECF = ?

(a) 300 (b) 600 (c) 750 (d) 800

Solution :(a) AC = AF ∠ACF = ∠AFC = x(Suppose) ; ∠ACD = 600 ∠FCD = (600 – x)

BF ‖ CE ∠AFC = ∠ECF ; x = 600 – x ⇒ 2x = 600 ⇒ x =

x = 3 00

Q 20. The length of the sides forming the right angle in a right angled triangle is ‘a’ and ‘b’. Three squares are inscribed outwards on the three sides of the triangle. What is the total sum of the area of the triangles and that of the squares so formed? (a) 2(a2 + b2) + ab (b) 2(a2 + b2) + 2.5 ab (c) 2(a2 + b2) + 0.5 ab (D) 2.5(a2 + b2)

Solution :(c) AB = a BC = b ; AC = √



⇒ Area of

; Area of square on sides AB = a2

Area of square on side BC = b2

Area of square on side AC = √

Total required area = a2 + b2 + b2 +

a2 = 2(a2 + b2) + 0.5 ab

Q 21. In ABC, ∠A = 90 and AD⊥ BC where D lies on BC. If BC = 8 cm, AC = 6 cm, then ABC : ACD = ? (a) 4 : 3 (b) 25 : 16 (c) 16 : 9 (d) 25 : 9 Solution :(c)

In ⊥ ⇒ Ratio of area of triangles = ratio of square of their corresponding

sides Hence,

6

Q 22. For a triangle, base is 6√ cm and two base angle are 30 and 60 . Then height of the triangle is

(a) 3√ cm (b) 4.5 cm (c) 4√ cm (d) 2√ cm Solution :(b)

Sin 30

⇒

√ ⇒ 3√3 ; Also, sin 60

⇒

√

√ ⇒ AD

√ √

4 5

Q 23. ABC is a right-angled triangle with AB = 6cm and BC = 8 cm. A circle with centre O has been inscribed inside ABC. The radius of the circle is (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm Solution :(b)



AC = √ √6 √36 64 √ ; OD = OE = OF = radii = r cm; Area of

[ = ⇒

6

6; ⇒ 3r + 4r + 5r = 24 ⇒ 12r =

24 ⇒r =

2

Q 24. If the radius of the circum circle of an equilateral triangle is 10 cm, then what will be the radius of the incircle inscribed in it? (a) 5 cm (b) 10 cm (c) 20 cm (d) 15 cm Solution :(c)

Ar (ABE) =

; OB : OE = 2 : 1

Ar (DOE) =

;

2

359.(a) Since the ratio of the radius of incircle and circumcircle of an equilateral 2

2 5

Q 25. In the given figure ∠ABC = 90

(a) 3 cm (b) 4 cm (c) 5 cm (d) 6 cm

Solution :(c) AC = √ 6 √ So, AM = MC = 5 cm. The midpoint of the diagonal AC is equidistant from the vertices A, B and C BM = AM = MC = 5 cm.



Q 26. A parallelogram ABCD and a rectangle ABEF are drawn between parallel lines EF and CD. If AB = 7cm and BE = 6.5 cm, then area of parallelogram will be (a) 22.75cm2 (b) 11.375cm2 (c) 45.5 cm2 (d) 45.0 cm2

Solution :(c) Area of rectangle ABEF = 7 6 5 45 5 Parallelogram ABCD and rectangle ABEF are based on the same base AB and between the same parallels AB ‖ CF)

45 5 cm2 Q 27. The area of 2. XY is drawn parallel to BC which divides AB in the ratio 3 : 5. If BY is joined then area of (a) 3.5 cm2 (b) 3.7 cm2 (c) 3.75 cm2 (d) 4.0 cm2

Solution :(c)

;

; Ar(AXY) =

2 25

5 2 25

3 3 5 2

Q 28. In the adjoining figure ABCD is a rectangle in which AE = EF = FB. Then the ratio of the areas of

(a) 1 : 4 (b) 1 : 6 (c) 2 : 5 (d) 2 : 3

Solution :(b) Let BC x and FB y EF AE ⇒ CD 3y Now ar

⇒ ar

2



; Area of rectangle ABCD = 3xy

( )

2

3

6 6

Q 29. In the adjoining figure the value of x is

(a) 6 cm (b) 7 cm (c) 6.7 cm (d) 7.7 cm

Solution :(c) PD ; (7 + x) 2 ; 7 + x =

; X = 13.7 – 7 = 6.7

Q 30. In the given figure O is the centre of the circle. If ∠AOC = 1400 then the value of ∠ABC is

(a) 1100 (b) 1200 (c) 1150 (d) 1300

Solution :(a)∠B = 1800 – 700 = 1100 Q 31. In the given figure O is the centre of the circle. If AB = 16 cm, CP = 6 cm, PD = 8 cm and AP > PB then value of AP is



(A) 12 cm (b) 24 cm (c) 8 cm (d) 6 cm Solution :(a) AP 6 6 x2 – 16x + 48 = 0.

2 4 2 4 Q 32. In the given figure O is the centre of the circle then value of x is

(a) 600 (b) 450 (c) 150 (d) 300

Solution :(d) In OB = OC ; ∠B = ∠C = 300 ; ∠ ∠ [ ; ∠D = 300

Q 33. In the given the value of x is

(a) 2.2 cm (b) 1.6 cm (c) 3 cm (d) 2.6 cm

Solution :(a) From the given figure, PT2 = PA ; 62 = 5(5 + x) ; 5 + x =

; x = 7.2 – 5 = 2.2 cm

Q 34. AB is the diameter of the circle and O is its centre. CD and AB intersect in such a way that OE = EB and CE = 6 cm, ED = 2 cm. Find the radius of the circle.



(a) 4 cm (b) 6 cm (c) 4√ cm (d) 8 cm Solution :(a) Let the radius of the circle be r 2 .

Now , AE (

)

6 2 4

Q 35. In the given figure A, B, C are points on the circumference of the circle and O is the centre. If ∠ABC is

(a) 600 (b) 750 (c) 900 (d) None of these

Solution :(b)∠AOC = 3600 – (900 + 1200) = 1500 ∠

∠

5 5

Q 36. In the given figure AD : DC = 2 : 3 then ∠ABC is

(a) 30 (b) 40 (c) 45 (d) 110

Solution :(b)

;

∠ ∠ 3 3 2 ; ∠ 2 ∠ 2 2 4 Q 37. In the given figure PQ is a tangent at point K, and LN is the diameter. If ∠KLN = 30 ∠



(a) 300 (b) 500 (c) 600 (d) 700

Solution :(c)∠LKN = 900 (angle of the semicircle) ∠ 1800 – (900 + 300) = 600

∠ ∠ 6 0 (angle in the alternate segment)

Q 38. In the given figure XY ‖AC, and XY divides the triangle into two equal parts. Then

(a)

(b)

√ (c)

√

√ (d)

√

√

Solution :(b) 2

2

√2 5

Q 39. In the given figure AM ⊥ BC and AN is the bisector of ∠A. Then ∠MAN will be- (If ∠B = 650, ∠C = 33

(a) 33

(b) 16

(c) 160 (d) 320

Solution :(c)∠MAN =

∠ ∠

65 33 6

Q 40. In the adjoining figure find the value of ∠QSR



(a) 500 (b) 650 (c) 700 (d) 750

Solution :(b) ∠PQO = ∠PRO = 900 [radii are perpendicular on tangents at the points of contact]

In Quadrilateral PQOR, ∠ROQ = 3600 – (900 + 900 + 500) = 3600 – 2300 = 1300

∠QSR =

∠QOR =

3 65

[Angle subtended by an arc on the remaining part of the circle is half the angle subtended on the centre.]

Practice Questions Geometry 500 Series Advance math 1. Point P is inside ∠BAC, if ∠BAC = 1150 & ∠PAC = 700 then find ∠BAP. (a) 700 (b) 1150 (c) 450 (d) None of these 2. If ∠AOB = 750, ∠BOC = 1050 then choose the correct answer: (a) OC ⊥ AB (b) OC ‖ OA (c) O, C & A are collinear (d) None of these 3. If point I is the incentre of ABC and ∠BIC = 1350, then (a) Acute angle (b) Equilateral triangle (c) Right angle triangle (d) Obtuse angle triangle 4. ABCD is a rhombus in which side AB = 4 cm and ∠ABC = 1200, then the length of the diagonal BD is (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm 5. In a ABC, ∠A + ∠B = 118 , ∠A + ∠C = 96 . Find the value of ∠A. (a) 36 (b) 40 (c) 30 (d) 34 6. In a ABC, ∠A = 3 ∠B = 70 and ∠B + ∠C = 130 , value of ∠A is (a) 20 (b) 50 (c) 110 (d) 30

7. In a triangle ABC, ∠A +

∠B + ∠C = 140 , then ∠B is

(a) 50 (b) 80 (c) 40 (d) 60 8. If I be the incentre of ABC and ∠B = 70 and ∠C = 50 , then the magnitude of ∠BIC is (a) 130 (b) 60 (c) 120 (d) 105 9. O is the in-centre of the ABC, if ∠BOC = 116 , then ∠BAC is (a) 42 (b) 62 (c) 58 (d) 52 10. In a triangle ABC, incentre is O and ∠BOC = 110 , then the measure of ∠BAC is: (a) 20 (b) 40 (c) 55 (d) 110 11. The circumcentre of a triangle ABC is O. If ∠BAC = 85 and ∠OAC =750 , then the value of ∠OAC is (a) 40 (b) 60 (c) 70 (d) 90 12. In ABC, D is the midpoint of BC. Length AD is 27 cm. N is a point in AD such that the length of DN is 12 cm. The distance of N from the centroid of ABC is equal to (a) 3 cm (b) 6 cm (c) 9 cm (d) 15 cm



13. The side BC of a triangle ABC is extended to D. If ∠ACD = 120 and ∠ABC =

∠CAB, then the value of ∠ABC is

(a) 80 (b) 40 (c) 60 (d) 20 14. In a ABC, If 2 ∠A = 3 ∠B = 6∠C, value of ∠B is (a) 60 (b) 30 (c) 45 (d) 90 15. A tree of hight ‘h’ metres is broken by a storm in such a way that its top touches the ground at a distance of ‘x’ metres from its root. Find the height at which the tree is broken. (Here h>x)

(a)

metre (b)

metre (c)

metre (d)

metre

16. In a ABC, the medians AD, BE and CF meet at G, then which of the following is true?

(a) AD + BE + CF >

(AB + BC + AC) (b) 2(AD + BE + CF) > (AB + BC + AC)

(c) 3(AD + BE + CF) > 4(AB + BC + AC) (d) 4(AD + BE + CF) >3(AB + BC + AC)

17. The internal bisectors of the angles B and C of a triangle ABC meet at I. If ∠BIC = ∠

+ X, then X is equal to

(a) 60 (b) 30 (c) 90 (d) 45 18. The measure of the angle between the internal and external bisectors of an angle is (a) 60 (b) 70 (c) 80 (d) 90 19. AD is the median of a triangle ABC and O is the centroid such that AO = 10 cm. Length of OD (in cm) is (a) 2 (b) 4 (c) 5 (d) 7 20. In a triangle ABC, median is AD and centroid is O. AO = 10 cm. The length of OD (in cm) is (a) 6 (b) 4 (c) 5 (d) 3.3

21. If in a triangle ABC, D and E are on the sides AB and AC, such that, DE is parallel to BC and

=

. If AC = 4 cm, then

AE is (a) 1.5 cm (b) 2.0 cm (c) 1.8 cm (d) 2.4 cm 22. O is the incentre of ABC and ∠A = 30 , then ∠BOC is (a) 100 (b) 105 (c) 110 (d) 90 23. The angles of a triangle are in the ratio 2 : 3 : 7. The measure of the smallest angle is (a) 30 (b) 60 (c) 45 (d) 90 24. If the angles of a triangle ABC are in the ratio 2 : 3 : 1, then the angles ∠A, ∠B and ∠C are (a) ∠A = 60 ∠B = 90 , ∠C = 30 (b) ∠A = 40 ∠B = 120 , ∠C = 20 (c) ∠A = 20 ∠B = 60 , ∠C = 60 (d) ∠A = 45 ∠B = 90 , ∠C = 45 25. In a ABC, D and E are two points on AB and AC respectively such that DE ‖ BC, DE bisects the ABC in two equal areas Then the ratio DB : AB is

(a) 1 : √2 (b) 1 : 2 (c) (√2 - 1) : √2 (d) √2 : 1 26. In triangle ABC a straight line parallel to BC intersects AB and AC at D and E respectively. If AB = 2AD then DE : BC is (a) 2 : 3 (b) 2 : 1 (c) 1 : 2 (d) 1 : 3 27. Two supplementary angles are in the ratio 2 : 3 The angles are (a) 33 , 57 (b) 66 , 114 (c) 72 , 108 (d) 36 , 54 28. In ABC, If AD⊥ BC, then AB2 + CD2 is equal to (a) 2 BD2 (b) BD2 + AC2 (c) 2 AC2 (d) None of these 29. If in a triangle ABC as drawn in the figure, AB = AC and ∠ACD = 120 , then ∠A is equal to

(a) 50 (b) 60 (c) 70 (d) 80



30. IF AD is the median of the triangle ABC and G be the centroid, then the ratio of AG : AD is (a) 1 : 3 (b) 2 : 1 (c) 3 : 2 (d) 2 : 3 31. BE and CF are two medians of ABC and G the centroid. FE cuts AG at O. If OG = 2 cm, then the length of AO is (a) 2 cm (b) 4 cm (c) 6 cm (d) 8 cm 32. Angle between the internal bisectors of two angle of a triangle ∠B and ∠C is 120 , then ∠A is (a) 20 (b) 30 (c) 60 (d) 90 33. The angle in a semi-circle is (a) a reflex angle (b) an obtuse angle (c) an acute angle (d) a right angle 34. In ABC, PQ is parallel to BC. If AP : PB = 1 : 2 and AQ = 3 cm; AC is equal to (a) 6 cm (b)9 cm (c) 12 cm (d) 8 cm 35. For a triangle ABC, D, E, F are the mid-points of its sides. If ABC = 24 sq. units then DEF is (a) 4 sq. units (b) 6 sq. units (c) 8 sq. units (d) 12 sq. units

36. For a triangle ABC, D and E are two points on AB and AC such that AD =

AB, AE =

AC. If BC = 12 cm, then DE

is (a) 5 cm (b) 4 cm (c) 3 cm (d) 6 cm 37. In ABC, DE ‖ AC D and E are two points on AB and CB respectively If AB cm and AD 4 cm then BE CE is (a) 2 : 3 (b) 2 : 5 (c) 5 : 2 (d) 3 : 2 38. In ABC, D and E are points on AB and AC respectively such That DE ‖ BC and DE divides the ABC into two parts of equal areas. Then ratio of AD and BD is

(a) 1 : 1 (b) 1 : √2 – 1 (c) 1 : √2 (d) 1 : √2 + 1 39. D is any point on side AC of ABC. If P, Q, X, Y are the midpoints of AB, BC, AD and DC respectively, then the ratio of PX and QY is (a) 1 : 2 (b) 1 : 1 (c) 2 : 1 (d) 2 : 3

40. The points D and E are taken on the side AB and AC of ABC such that AD =

AB, AE =

AC. If the length of BC is 15

cm, then the length of DE is: (a) 10 cm (b) 8 cm (c) 6 cm (d) 5 cm 41. The external bisector of ∠B and ∠ of ABC (where AB and AC extended to E and F respectively) meet at point P. If ∠BAC = 100 , then the measure of ∠BPC (a) 50 (b) 80 (c) 40 (d) 100 42. The equidistant point from the vertices of a triangle is called its: (a) Centriod (b) In centre (c) Circum centre (d) Ortho centre 43. If in ABC, ∠ABC = 5 ∠ACB and ∠BAC = 3 ∠ACB, then ∠ABC = ? (a) 130 (b) 80 (c) 100 (d) 120 44. Let O be the in-centre of a triangle ABC and D be a point on the side BC of ABC, such that OD ⊥ BC. If ∠BOD = 15 , then ∠ABC = (a) 75 (b) 40 (c) 150 (d) 110 45. Inside a triangle ABC, a straight line parallel to BC intersect AB and AC at the point P and Q respectively. If AB = 3 PB, then PQ : BC is (a) 1 : 3 (b) 3 : 4 (c) 1 : 2 (d) 2 : 3 46. AD is the median of a triangle ABC and O is the centroid such that AO = 10 cm. The length of OD (in cm) is (a) 4 (b) 5 (c) 6 (d) 8 47. In ABC, the internal bisectors of ∠ABC and ∠ACB meet at I and ∠BAC= 500 . The measure of ∠BIC is (a) 1050 (b) 1150 (c)1250 (d) 1300 48. If AD, BE and CF are medians of ABC, then which one of the following, statements is correct? (a) (AD + BE + CF) < AB + BC + CA (b) AD + BE + CF > AB + BC + CA

(c) AD + BE + CF = AB + BC + CA (d) AD + BE + CF = √2(AB + BC + CA) 49. The exterior angles obtained on producing the base BC of a triangle ABC in both ways are 120 and 105 , then the vertical ∠A of the triangle is of measure



(a) 36 (b) 40 (c) 45 (d) 55 50. The perpendiculars drawn from the vertices to the opposite sides of a triangle, meet at the point whose name is (a) incentre (b) Circumcentre (c) centroid (d) orthocentre 51. In a triangle ABC, AB + BC = 12 cm, BC + CA = 14 cm and CA + AB = 18 cm. Find the radius of the circle (in cm) which has the same perimeter as the triangle.

(a)

(b)

(c)

(d)

52. In a ABC, AB = BC, ∠B = x and ∠A = (2x – 20) . The ∠B is (a) 54 (b) 30 (c) 40 (d) 44 53. If each interior angle of a regular polygon is 1080 ten find its no. of sides. (a) 5 (b) 6 (c) 8 (d) 7 54. In any ABC, Bisectors of ∠B and ∠C meet at point O. ∠A = 700 then ∠BOC = ? (a) 1250 (b) 1150 (c) 1600 (d) 1400 55. ABCD is a cyclic trapezium in which AD ‖ BC. If ∠ABC = 700, then the measure of ∠BCD is (a) 600 (b) 700 (c) 400 (d) 800 56. ABCD is a rectangle & ABP is a right angled ∠PAB = 600 then find ∠PDC. (a) 150 (b) 200 (c) 300 (d) 100

57. In ABC ∠A + ∠B = 65 , ∠B + ∠C = 140 , then find ∠B. (a) 40 (b) 25 (c) 35 (d) 20

58. In a triangle ABC, ∠A = 90 . ∠C = 55 , ⊥ . What is the value of ∠BAD ? (a) 35 (b) 60 (c) 45 (d) 55 59. I is the incentre of ABC. If ∠ABC = 60 , ∠BCA = 80 , the the ∠BIC is (a) 90 (b) 100 (c) 110 (d) 120 60. O is the centre and arc ABC subtends an angle of 130 at O, AB is extended to P. Then ∠PBC is (a) 75 (b) 70 (c) 65 (d) 80

61. In triangle ABC, ∠BAC = 75 , ∠ABC = 45 . is produced to D. If ∠ACD = x , then

% of 60 is

(a) 30 (b) 48 (c) 15 (d) 90 62. If O be the circumcentre of a triangle PQR and ∠QOR = 110 , ∠OPR = 25 , then the measure of ∠PRQ is (a) 65 (b) 50 (c) 55 (d) 60 63. If ABC is similar to DEF, such that ∠A = 47 and ∠E = 63 the ∠C is equal to: (a) 40 (b) 70 (c) 65 (d) 37 64. In ABC, ∠B = 60 and ∠C = 40 . If AD and AE be respectively the internal bisector of ∠A and perpendicular on BC, then the measure of ∠DAE is (a) 5 (b) 10 (c) 40 (d) 60

65. In ABC ∠A = ∠B = 60 , AC = √ 3 cm. The lines AD and BD intersect at D with ∠D = 90 . If DB = 2 cm, then the length of AD is (a) 3 cm (b) 3.5 cm (c) 4 cm (d) 4.7 cm 66. Ashok has drawn an angle of measure 45 27’when he was asked to draw an angle of 45 . The percentage error in his drawing is (a) 0.5% (b) 1.0% (c) 1.5% (d) 2.0% 67. ∠A, ∠B, ∠C are three angles of a triangle. If ∠A - ∠B = 15 , ∠B - ∠C = 30 , then ∠A, ∠B and ∠C are (a) 80 , 60 , 40 (b) 70 , 50 , 60 (c) 80 , 65 , 35 (d) 80 , 55 , 45 68. All sides of a quadrilateral ABCD touch a circle. If AB = 6 cm, BC = 7.5 cm, CD = 3 cm, then DA is (a) 3.5 cm (b) 4.5 cm (c) 2.5 cm (d) 1.5 cm 69. D is a point on the side BC of a triangle ABC such that AD ⊥ BC, E is a point on AD for which AE : ED = 5 : 1. If ∠BAD = 30 and tan (∠ACB) = 6 tan(∠DBE), then ∠ACB = (a) 30 (b) 45 (c) 60 (d) 15 70. In a ABC ∠A : ∠B : ∠C = 2 : 3 : 4. A line CD drawn ‖ to AB, then the ∠ACD is: (a) 40 (b) 60 (c) 80 (d) 20 71. In a triangle ABC, BC is produced to D so that CD = AC. If ∠BAD = 111 and ∠ACB = 80 , then the measure of ∠ABC is:



(a) 31 (b) 33 (c) 35 (d) 29 72. In the following figure, AB be diameter of a circle whose centre is O. If ∠AOE = 150 , ∠DAO = 51 then the measure of ∠CBE is:

(a) 115 (b) 110 (c) 105 (d) 120 73. In a ABC, AB = AC and BA is produced to D such that AC = AD. Then the ∠BCD is (a) 100 (b) 60 (c) 80 (d) 90 74. Two chords AB, CD of a circle with centre O intersect each other at P. ∠ADP = 23 and ∠APC = 70 then the ∠BCD is (a) 45 (b) 47 (c) 57 (d) 67 75. In triangle PQR, points A, B and C are taken on PQ, PR and QR respectively such that QC = AC and CR = CB. If ∠QPR = 40 then ∠ACB is equal to: (a) 140 (b) 40 (c) 70 (d) 100 76. Internal bisectors of angles ∠B and ∠C of a triangle ABC meet at O. If ∠BAC = 80 , then the value of ∠BOC is (a) 120 (b) 140 (d) 110 (d) 130 77. In ABC, draw BE ⊥ AC and CF ⊥ AB and the perpendicular BE and CF intersect at the point O. If ∠BAC = 70 , then the value of ∠BOC is (a) 125 (b) 55 (c) 150 (d) 110 78. The angle between the external bisectors of two angles of a triangle is 60 . Then the third angle of the triangle is (a) 40 (b) 50 (c) 60 (d) 80 79. Internal bisectors of ∠B and ∠C of ABC intersect at O. If ∠BOC = 102 , then the value of ∠BAC is (a) 12 (b) 24 (c) 48 (d) 80 80. A circle (with centre at O) is touching two intersecting lines AX and BY. The two points of contact A and B subtend an angle of 65 at any point C on the circumference of the circle. If P is the point of intersection of the two lines, then the measure of ∠APO is (a) 25 (b) 65 (c) 90 (d) 40 81. The internal bisectors of ∠ABC and ∠ACB of ABC meet each other at O. If ∠BOC = 110 , then ∠BAC is equal to (a) 40 (b) 55 (c) 90 (d) 110 82. ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 and ∠BAC = 30 , then ∠BCD is: (a) 75 (b) 90 (c) 120 (d) 60 83. Two chords AB and CD of a circle with centre O intersect each other at the point P. If ∠AOD = 20 and ∠BOC = 30 , then ∠BPC is equal to: (a) 50 (b) 20 (c) 25 (d) 30 84. A straight line parallel to BC of ABC intersects AB and AC at points P and Q respectively. AP = QC, PB = 4 units and AQ = 9 units, then the length of AP is: (a) 25 units (b) 3 units (c) 6 units (d) 6.5 units 85. Two chords AB and CD of circle whose centre is O, meet at the point P and ∠AOC = 50 , ∠BOD = 40 . Then the value of ∠BPD is (a) 60 (b) 40 (c) 45 (d) 75 86. Two line segments PQ and RS intersect at X in such a way that XP = XR, If ∠PSX = ∠RQX, then one must have (a) PR = QS (b) PS = RQ (c) ∠XSQ = ∠XRP (d) ar( PXR) = ar( QXS)

87. In a ABC, 2 + 2 = 2 and = √2 ̅̅ ̅̅ then ∠ABC is: (a) 30 (b) 45 (c) 60 (d) 90 88. In ABC, ∠A + ∠B = 145 and ∠C + 2∠B = 180 . State which one of the following relations is true? (a) CA = AB (b) CA < AB (c) BC > AB (d) CA > AB



89. In a ABC, ∠A = 600. If the perpendiculars drawn from vertices B & C to front sides intersect each other at P then find ∠BPC. (a) 1480 (b) 1200 (c) 1380 (d) 1420 90. The exterior angle of a cyclic quadrilateral is 500. The measure of its opposite internal angle is (a) 1300 (b) 400 (c) 500 (d) 900 91. In the following figure ∠B = 380, AC = BC and AD = CD then ∠D is

(a) 260 (b) 280 (c) 380 (d) 520 92. Point O is the circumcentre of ABC and ∠OBC = 350 then ∠BAC is (a) 550 (b) 1100 (c) 700 (d) 350

93. In a PQR, ∠RPQ = 90 , = 6 cm and = 8 cm, then the radius of the circum circle of PQR is (a) 5 cm (b) 3 cm (c) 4 cm (d) 4.5 cm 94. O is the circumcentre of ABC, given ∠BAC = 85 and ∠BCA = 55 , find ∠OAC. (a) 40 (b) 50 (c) 60 (d) 80 95. The radius of two concentric circles are 17 cm and 10 cm. A straight line ABCD intersects the larger circle at the point A and D and intersects the smaller circle at the points B and C. If BC = 12 cm, then the length of AD (in cm) is: (a) 20 (b) 24 (c) 30 (d) 34 96. P and Q are centre of two circles with radii 9 cm and 2 cm respectively, where PQ = 17 cm. R is the centre of another circle of radius x cm, which touches each of the above two circles externally. If ∠PRQ = 90 , then the value of x is (a) 4 cm (b) 6 cm (c) 7 cm (d) 8 cm 97. If the circum radius of an equilateral triangle ABC be 8 cm, then the height of the triangle is (a) 16 cm (b) 6 cm (c) 8 cm (d) 12 cm 98. ABC is an equilateral triangle and O is its circum centre, then the ∠AOC is (a) 100 (b) 110 (c) 120 (d) 130 99. Triangle PQR circumscribes a circle with centre O and radius r cm such that ∠PQR = 90 . If PQ = 3 cm, QR = 4 cm, then the value of r is: (a) 2 (c) 1.5 (c) 2.5 (d)1 100. If the ABC is right angled at B, find its circum radius if the sides AB and BC are 15 cm and 20 cm respectively. (a) 25 cm (b) 20 cm (c) 15 cm (d) 12.5 cm 101. The radius of the circum circle of a right angled triangle is 15 cm and the radius of its inscribed circle is 6 cm. Find the sides of the triangle. (a) 30, 40, 41 (b) 18, 24, 30 (c) 30, 24, 25 (d) 24, 36, 20 102. I and O are respectively the in – centre and circum centre of a triangle ABC. The line AI produced intersects the

circum circle of ABC at the point D. If ∠ABC = x , ∠BID = y and ∠BOD = z , then

=

(a) 3 (b) 1 (c) 2 (d) 4 103. The length of radius of a circum circle of a triangle having sides 3 cm, 4 cm and 5 cm is: (a) 2 cm (b) 2.5 cm (c) 3 cm (d) 1.5 cm 104. The length of the two side forming the right angle of a right – angled triangle are 6 cm and 8 cm. The length of its circum – radius is: (a) 5 cm (b) 7 cm (c) 6 cm (d) 10 cm 105. AC is the diameter of a circum circle of ABC. Chord ED is parallel to the diameter AC. If ∠ CBE = 50 , then the measure of ∠DEC is (a) 50 (b) 90 (c) 60 (d) 40



106. P is point inside two parallel lines AB & CD. If ∠ABP = 300, ∠CDP = 450 then find ∠DPB. (a) 850 (b) 450 (c) 300 (d) 750 107. Any angle is 160 larger than its complement, then find the value of that angle. (a) 370 (b) 530 (c) 430 (d) 470 108. The ratio of areas of two isosceles triangles is 9 : 16. If the corresponding angles are equal then find the ratio of altitudes. (a) 4 : 3 (b) 3 : 4 (c) 81 : 256 (d) None of these 109. ABCD is a cyclic trapezium in which AD ║ BC. If ∠ABC = 72 then the value of ∠ BCD is (a) 162 (b) 18 (c) 108 (d) 72 110. The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and the bigger circle at E. Point A is joined to D. The length of AD is (a) 20 cm (b) 19 cm (c) 18 cm (d) 17 cm 111. The distance between the centres of two circles with radii 9 cm and 16 cm is 25 cm. The length of the segment of the tangent between them is

(a) 24 cm (b) 25 cm (c)

cm (d) 12 cm

112. Two circles are of radii 7 cm and 2 cm their centres being 13cm apart. Then the length of direct common tangent to the circles between the points of contact is (a) 12 cm (b) 15 cm (c) 10 cm (d) 5 cm 113. The radius of a circle is 6 cm. The distance of a point lying outside the circle from the centre is 10 cm. The length of the tangent drawn from the outside point to the circle is (a) 5 cm (b) 6 cm (c) 7 cm (d) 8 cm 114. The radius of two concentric circles are 9 cm and 15 cm. If the chord of the greater circle be a tangent to the smaller circle, then the length of that chord is (a) 24 cm (b) 12 cm (c) 30 cm (d) 18 cm 115. PQ is a chord of length 8 cm, of a circle with centre O and of radius 5 cm. The tangents at P and Q intersect at a point T. The length of TP is

(a)

cm (b)

cm (c)

cm (d)

cm

116. If PA and PB are two tangents to a circle with centre O such that ∠AOB = 110 , then ∠APB is (a) 90 (b) 70 (c) 60 (d) 55 117. ST is a tangent to the circle at P and QR is a diameter of the circle. If ∠RPT = 50 , then the value of ∠SPQ is (a) 40 (b) 60 (c) 80 (d) 100 118. DE is a tangent to the circum circle of ABC at the vertex A such that DE ‖ BC. If AB = 17 cm, then the length of AC is equal to (a) 16.0 cm (b) 16.8 cm (c) 17.3 cm (d) 17 cm 119. The radii of two circles are 5 cm and 3 cm, the distance between their centre is 24 cm. Then the length of the transverse common tangent is

(a) 16 cm (b) 15√2 cm (c) 16√2 cm (d) 15 cm

120. The length of a tangent from an external point to a circle is 5√3 unit. If radius of the circle is 5 units, then the distance of the point from the circle is (a) 5 units (b) 15 units (c) -5 units (d) -15 units 121. If the radii of two circles be 6 cm and 3 cm and the length of the transverse common tangent be 8 cm, then the distance between the two centres is

(a) √ 45 (b) √ 4 (c) √ 5 (d) √ 35 122. The length of the tangent drawn to a circle of radius 4 cm from a point 5 cm away from the cnetre of the circle is

(a) 3 cm (b) 4√2 cm (c) 5√2 (d) 3√2 cm 123. The minimum number of common tangents drawn to two circles when both the circles touch each other externally is (a) 1 (b) 2 (c) 3 (d) 0



124. From a point P, two tangents PA and PB are drawn to a circle with centre O. If OP is equal to diameter of the circle, then ∠APB is (a) 45 (b) 90 (c) 30 (d) 60 125. Two circles intersect at A and B. P is a point on produced BA. Pt and PQ are tangents to the circles. The relation of PT and PQ is (a) PT = 2PQ (b) PT < PQ (c) PT > PQ (d) PT = PQ 126. P and Q are two points on a circle with centre at O. R is a point on the minor are of the circle, between the points P and Q. The tangents to the circle at the points P and Q meet each other at the point S. If ∠PSQ = 20 , then ∠PRQ = ? (a) 80 (b) 200 (c) 160 (d) 100

127. PR is tangent to a circle, with centre O and radius 4 cm, at point Q. If ∠POR = 90 , OR = 5 cm and OP =

cm, then

(in cm) the length of PR is:

(a) 3 (b)

(c)

(d)

128. The distance between the centre of two equal circles. Each of radius 3 cm, is 10 cm. The length of a transverse common tangent is (a) 8 cm (b) 10 cm (c) 4 cm (d) 6 cm 129. Two circles touch each other externally at P. AB is a direct common tangent to the two circles, A and B are point of contact and ∠PAB = 35 . Then ∠ABP is (a) 35 (b) 55 (c) 65 (d) 75 130. If a chord of a circle of radius 5 cm is a tangent to another circle of radius 3 cm, both the circles being concentric, then the length of the chord is (a) 10 cm (b) 12.5 cm (c) 8 cm (d) 7 cm 131. Two circles touch each other externally at point A and PQ is a direct common tangent which touches the circles at P and Q respectively. Then ∠PAQ = (a) 45 (b) 90 (c) 80 (d) 100 132. The tangents at two points A and B on the circle with centre O intersects at P; If in quadrilateral PAOB, ∠AOB : ∠APB = 5 : 1, then measure of ∠APB is: (a) 30 (b) 60 (c) 45 (d) 15 133. AB is a chord to a circle and PAT is the tangent to the circle at A. If ∠BAT = 75 and ∠BAC = 45 , C being a point on the circle, then ∠ABC is equal to (a) 40 (b) 45 (c) 60 (d) 70 134. The tangents are drawn at the extremities of diameter AB of a circle with centre P. If a tangent circle with circle at the point C intersects the other two tangents at Q and R, then the measure of the ∠QPR is (a) 45 (b) 60 (c) 90 (d) 180 135. Point O is the circum centre of the triangle ABC. If ∠BAC = 85 and ∠BCA = 75 then the value of ∠OAC is (a) 40 (b) 60 (c) 70 (d) 90 136. In AD is angle bisector of ∠A. If AB = 8 cm, AC = 10 cm & BC = 13.5 cm then find BD. (a) 8 cm (b) 7.5 cm (d) 6 cm (d) 6.5 cm 137. In an equilateral incentre, circumcentre and orthocentre are (a) Collinear (b) Coincident (c) Circular (d) Not related 138. What will be the value of x.

(a) 150 (b) 22.50 (c) 200 (d) 300



139. In ABC, ∠ABC = 70 , ∠BCA = 40 . O is the point of intersection of the perpendicular bisectors of the sides, then the angle ∠BOC is (a) 100 (b) 120 (c) 130 (d) 140 140. In the given figure, ∠ONY = 50 and ∠OMY = 15 , Then the value of the ∠MON is

(a) 30 (b) 40 (c) 20 (d) 70 141. In a circle of radius 21 cm, an arc subtends an angle of 72 at the centre. The length of the arc is (a) 21.6 cm (b) 26.4 cm (c) 13.2 cm (d) 19.8 cm 142. Two chords of lengths a metre and b metre sub end angles 60 and 90 at the centre of the circle respectively. Which of the following is true?

(a) b = √2 a (b) a = √2 b (c) a=2b (d) b= 2a 143. The angle subtended by a chord at its centre is 60 , then the ratio between chord and radius is

(a) 1 : 2 (b) 1 : 1 (c) √2 : 1 (d) 2 : 1 144. A chord 12 cm long is drawn in a circle of diameter 20 cm. The distance of the chord from the centre is (a) 8 cm (b) 6 cm (c) 10 cm (d) 16 cm 145. Two circles with centres A and B of radii 5 cm and 3 cm respectively touch each other internally. If the perpendicular bisector of AB meets the bigger circle of AB meets the bigger circle in P and Q, then the value of PQ is

(a) √6 cm (b) 2√6 cm (c) 3√6 cm (d) 4√6 cm 146. The length of the common chord of two circles of radii 30 cm and 40 cm whose centres are 50 cm apart, is (in cm) (a) 12 (b) 24 (c) 36 (d) 48 147. Two circles having radii r units intersect each other in such a way that each of them passes through the centre of the other. Then the length of their common chord is

(a) √2r units (b) √3 r units (c) √5 r units (d) r units 148. Two circles C1 and C2 touch each other internally at P. Two lines PCA and PDB meet the circles C1 In C, D and C2 in A, B respectively. If ∠BDC = 120 , then the value of ∠ABP is equal to (a) 60 (b) 80 (c) 100 (d) 120 149. Two circles intersect each other at the points A and B, A straight line parallel to AB intersects the circles at C, D, E and F. If CD = 4.5 cm, then the measure of EF is (a) 1.50 cm (b) 2.25 cm (c) 4.50 cm (d) 9.00 cm 150. Two circles touch externally at P. QR is a common tangent of the circles touching the circles at Q and R. Then measure of ∠QPR is (a) 60 (b) 30 (c) 90 (d) 45 151. Chords AC and BD of a circle with centre O intersect at right angles at E. If ∠ OAB = 25 , then the value of ∠EBC is (a) 30 (b) 25 (c) 20 (d) 15 152. In a circle of radius 17 cm, two parallel chords of length 30 cm and 16 cm are drawn. If both the chords are on the same side of the centre, then the distance between the chords is (a) 9 cm (b) 7 cm (c) 23 cm (d) 11 cm 153. Two chords AB and CD of a circle with centre O, intersect each other at P. If ∠AOD = 100 and ∠BOC = 70 , then the value of ∠APC is (a) 80 (b) 75 (c) 85 (d) 95



154. Two parallel chords are drawn in a circle of diameter 30 cm. The length of one chord is 24 cm and the distance between the two chords is 21 cm. The length of the other chord is (a) 10 cm (b) 18 cm (c) 12 cm (d) 16 cm

155. A, B, C are three points on the circumference of a circle and if = = 5√2 cm and ∠BAC = 90 , find the radius. (a) 10 cm (b) 5 cm (c) 20 cm (d) 15 cm 156. If two concentric circles are of radii 5 cm and 3 cm, then the length of the chord of the large circle which touches the smaller circle is (a) 6 cm (b) 7 cm (c) 10 cm (d) 8 cm

157. ‘O’ is the centre of the circle, AB is a chord of the circle, OM ⊥ AB. If AB = 20 cm, OM = 2√ cm, then radius of the circle is (a) 15 cm (b) 12 cm (c) 10 cm (d) 11 cm 158. In a right angled triangle, the circumcentre of the triangle lies (a) inside the triangle (b) outside the triangle (c) on midpoint of the hypotenuse (d) on one vertex 159. The three equal circles touch each other externally. If the centres of these circles be A, B, C then ABC is (a) a right angle triangle (b) an equilateral triangle (c) an isosceles triangle (d) a scalene triangle 160. For a triangle circumcentre lies on one of its sides. The triangle is (a) right angled (b) obtuse angled (c) isosceles (d) equilateral 161. Each of the circles of equal radii with centres A and B pass through the centre of one another circle they cut at C and D then ∠DBC is equal to (a) 60 (b) 100 (c) 120 (d) 140 162. If the chord of a circle is equal to the radius of the circle, then the angle subtended by the chord at a point on the minor arc is (a) 150 (b) 60 (c) 120 (d) 30 163. A, B, C, D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130 . And ∠ECD = 20 . ∠BAC is (a) 120 (b) 90 (c) 100 (d) 110 164. N is the foot of the perpendicular from a point P of circle with radius 7 cm, on a diameter AB of the circle. If the length of the chord PB is 12 cm, the distance of the point N from the point B is

(a) 6

cm (b) 12

cm (c) 3

cm (d) 10

cm

165. Two circles touch each other internally. Their radii are 2 cm and 3 cm. The biggest chord of the greater circle which is outside the inner circle is of length

(a) 2√2 (b) 3√2 (c) 2√3 (d) 4√2 166. A, B and C are the three points on a circle such that the angles subtended by the chords AB and AC at the centre O are 90 and 110 respectively. ∠BAC is equal to (a) 70 (b) 80 (c) 90 (d) 100 167. Two circles of same radius 5 cm, intersect each other at A and B if AB = 8 cm, then the distance between the centre is: (a) 6 cm (b) 8cm (c) 10 cm (d) 4 cm 168. Two circles touch each other externally. The distance between their centre is 7 cm. If the radius of one circle is 4 cm. then the radius of the other circle is (a) 3.5 cm (b) 3 cm (c) 4 cm (d) 2 cm 169. A square ABCD is inscribed in a circle of unit radius. Semicircles are described on each side as a diameter. The area of the region bounded by the four semicircles and the circle is (a) 1 sq. unit (b) 2 sq. unit (c) 1.5 sq. unit (d) 2.5 sq. unit 170. AB is the chord of a circle with centre O and DOC is a line segment originating from a point D on the circle and intersecting AB produced at C such that BC = OD. If ∠BCD = 20 , then ∠AOD = ? (a) 20 (b) 30 (c) 40 (d) 60 171. The length of two chords AB and AC of a circle are 8 cm and 6 cm and ∠BAC = 90 , then the radius of circle is (a) 25 cm (b) 20 cm (c) 4 cm (d) 5 cm



172. Two circles with centre P and Q intersect at B and C. A, D are points on the circle such that A, C, D are collinear. If ∠APB = 130 , and ∠BQD = x , then the value of x is (a) 65 (b) 130 (c) 195 (d) 135 173. AB = 8 cm and CD = 6 cm are two parallel chords on the same side of the centre of a circle. The distance between them is 1 cm. The radius of the circle is (a) 5 cm (b) 4 cm (c) 3 cm (d) 2 cm 174. Chords AB and CD of a circle intersect at E and are perpendicular to each other. Segments AE, EB and ED are of lengths 2 cm, 6 cm and 3 cm respectively. Then the length of the diameter of the circle (in cm) is

(a) √65 (b)

√65 (c) 65 (d)

175. AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If the chords are on the opposite sides of the centre and distance between them is 17 cm, then the radius of the circle is: (a) 11 cm (b) 12 cm (c) 13 cm (d) 10 cm 176. The distance between two parallel chords of length 8 cm each in a circle of diameter 10 cm is (a) 6 cm (b) 7 cm (c) 8 cm (d) 5.5 cm

177. AB and CD are two parallel chords on the opposite sides of the centre of the circle. If = 10 cm, = 24 cm and the radius of the circle is 13 cm, the distance between the chords is (a) 17 cm (b) 15 cm (c) 16 cm (d) 18 cm 178. A circle (with centre at O) is touching two intersecting lines AX and BY. The two points of contact A and B subtend an angle of 65 at any point C on the circumference of the circle. If P is the point of intersection of the two lines, then the measure of ∠APO is (a) 25 (b) 65 (c) 90 (d) 40 179. Chords AB and CD of a circle intersect externally at P. If AB = 6 cm, CD = 3 cm and PD = 5 cm, then the length of PB is (a) 5 cm (b) 7.35 cm (c) 6 cm (d) 4 cm 180. One chord of a circle is known to be 10.1 cm. The radius of this circle must be: (a) 5 cm(b) greater than 5 cm (c) greater than or equal to 5 cm (d) less than 5 cm 181. If two equal circles whose centres are O and O’, intersect each other at the point A and B, OO’= 12 cm and AB = 16 cm, then the radius of the circle is (a) 10 cm (b) 8 cm (c) 12 cm (d) 14 cm 182. The length of the chord of a circle is 8 cm and perpendicular distance between centre and the chord is 3 cm. Then the radius of the circle is equal to: (a) 4 cm (b) 5 cm (c) 6 cm (d) 8 cm 183. A unique circle can always be drawn through x number of given non-collinear points, then x must be: (a) 2 (b) 3 (c) 4 (d) 1 184. The length of the common chord of two intersecting circles is 24 cm. If the diameter of the circles are 30 cm and 26 cm, then the distance between the centre (in cm) is (a) 13 (b) 14 (c) 15 (d) 16 185. Two equal circles of radius 4 cm intersect each other such that each passes through the centre of the other. The length of the common chord is:

(a) 2√3 cm (b) 4√3 cm (c) 2√2 cm (d) 8 cm 186. The length of a chord of a circle is equal to the radius of the circle. The angle which this chord subtends in the major segment of the circle is equal to (a) 30 (b) 45 (c) 60 (d) 90

187. A chord AB of circle C1 of radius (√3 ) cm touches a circle c2 which is concentric to c1. If the radius of c2is

(√3 ) cm., the length of AB is:

(a) 2√3

cm (b) 8√3 (c) 4√3

cm (d) 4√3 188. If O is the centre of circle then ∠BOD = ?.



(a) 1400 (b) 1800 (c) 1000 (d) 1200 189. If in the given figure AB ‖ DC then find the value of ∠ADC.

(a) 700 (b) 600 (c) 450 (d) 750 190. In a ABC, AD is bisector of ∠A. If BD = 2.5 cm, AC = 4.2 cm, AB = 6 cm then DC = ? (a) 2.1 (b) 3 cm (c) 1.2 cm (d) None of these 191. Two circles touch each other internally. If their radii are 2 cm & 3 cm. Find the length of that chord of larger circle which touches the smaller one.

(a) 2√2 cm (b) 4√2 cm (c) 3√2 cm (d) None of these 192. ABCD is a rhombus whose side AB = 4 cm and ∠ABC = 120 , then the length of diagonal BD is equal to: (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm

193. ABCD is a trapezium whose side parallel to . Diagonal and intersect at O. If = 3, = x – 3,

= 3x – 19 and = x – 5, the value (s) of x will be: (a) 7, 6 (b) 12, 6 (c) 7, 10 (d) 8, 9 194. ABCD is a rhombus. AB is produced to F and BA is produced to E such that AB = AE= BF. Then: (a) ED > CF (b) ED ⊥CF (c) ED2 + CF2 = EF2 (d) ED ‖ CF 195. The ratio of the angles ∠A and ∠B of a non-square rhombus ABCD is 4 : 5, then the value of ∠C is: (a) 50 (b) 45 (c) 80 (d) 95 196. In a quadrilateral ABCD, with unequal sides if the diagonals AC and BD intersect at right angles. Then (a) AB2 + BC2 = CD2 + DA2 (b) AB2 + CD2 = BC2 + DA2 (c) AB2 + AD2 = BC2 + CD2 (d) AB2 + BC2 = 2(CD2 +DA2)

197. ABCD is a rhombus. A straight line through C cuts AD produced at P and AB produced at Q. If DP =

AB, then the

ratio of the length of BQ and AB is (a) 2 : 1 (b) 1 : 2 (c) 1 : 1 (d) 3 : 1 198. If PAB is an intersecting line which intersects the circle at points A & B. PT is a tangent then PA will be (a) PT2 (b) PT (c) AT (d) None of these 199. AB is a diameter of a circle point P is on the circle. If ∠PAB = 400 then find ∠PBA. (a) 400 (b) 600 (c) 500 (d) None of these



200. If base and height of a be reduced to half then find the ratio of areas of previous & new triangles (a) 4 : 1 (b) 2 : 1 (c) 1 : 4 (d) None of these 201. The intersection point of angle bisectors of a is known as (a) incentre (b) circumcentre (c) orthocentre (d) None of these 202. ABCD is a cyclic trapezium such that AD ‖ BC, if ∠ABC = 70 , then the value of ∠BCD is: (a) 60 (b) 70 (c) 40 (d) 80 203. If an exterior angle of a cyclic quadrilateral be 50 , then the interior opposite angle is: (a) 130 (b) 40 (c) 50 (d) 90 204. If ABCD be a cyclic quadrilateral in which ∠A = 4x ∠B = 7x , ∠C = 5y , ∠D = y , then x : y is (a) 3 : 4 (b) 4 : 3 (c) 5 : 4 (d) 4 : 5 205. A quadrilateral ABCD circumscribes a circle and AB = 6 cm, CD = 5 cm and AD = 7 cm. The length of side BC is (a) 4 cm (b) 5 cm (c) 3 cm (d) 6 cm 206. ABCD is a cyclic quadrilateral and AD is a diameter. If ∠DAC = 55 then value of ∠ABC is (a) 55 (b) 35 (c) 145 (d) 125 207. In a cyclic quadrilateral ∠A + ∠ = ∠B + ∠D =?

(a) 270 (b) 360 (c) 90 (d) 180 208. ABCD is a cyclic quadrilateral. The side AB is extended to E in such a way the BE = BC, If ∠ADC = 70 , ∠BAD = 95 , then ∠DCE is equal to (a) 140 (b) 120 (c) 165 (d) 110 209. In a cyclic quadrilateral ABCD m∠A + m∠B + m∠C + m∠D =? (a) 90 (b) 360 (C) 180 (d) 120 210. The diagonals AC and BD of a cyclic quadrilateral ABCD intersect each other at the point P. Then, it is always true that (a) BP. AB = CD. CP (b) AP. CP = BP. DP (c) AP. BP = CP. DP (d) AP. CD = AB. CP 211. A cyclic quadrilateral ABCD is such that AB = BC, AD = DC, AC ⊥ BD, ∠CAD = , Then the angle ∠ABC =

(a) (b)

(c) 2 (d) 3

212. ABCD is a cyclic quadrilateral, AB and DC are produced to meet at P. If ∠ADC = 70 and ∠DAB = 60 , then the ∠PBC + ∠PCB is (a) 130 (b) 150 (c) 155 (d) 180 213. ABCD is a cyclic trapezium with AB ‖ DC and AB = diameter of the circle. If ∠CAB = 30 , then ∠ADC is (a) 60 (b) 120 (c) 150 (d) 30 214. ABCD is a cyclic quadrilateral and O is the centre of the circle. If ∠COD =140 ∠BAC = 40 , then the value of ∠BCD is equal to (a) 70 (b) 90 (c) 60 (d) 80 215. ABCD is a cyclic trapezium whose sides AD and BC are parallel to each other. If ∠ABC = 72 , then the measure of the ∠BCD is (a) 162 (b) 18 (c) 108 (d) 72 216. ABCD is a cyclic parallelogram. The angle ∠B is equal to: (a) 30 (b) 60 (c) 45 (d) 90 217. The perpendicular bisectors of sides of a passes through: (a) incentre (b) circumcentre (c) centroid (d) None of these 218. A chord of circle with radius 58 cm is the tangent to a circle with radius 3 cm long. If these two circles are concentric, then the length of that chord is



(a) 10 cm (b) 12.5 cm (c) 8 cm (d) 7 cm 219. A shape formed after joining the respective mid-points of a quadrilateral is: (a) Rhombus (b) Square (c) Rectangle (d) None of these 220. In □ABCD if AB = BC = CD = CA and AC BD, then this quadrilateral is a: (a) Trapezium (b) Square (c) Rhombus (d) None of these 221. The side AB of a parallelogram ABCD is produced to E in such way that BE = AB. DE intersects BC at Q. The point Q dives BC in the ratio (a) 1 : 2 (b) 1 : 1 (c) 2 : 3 (d) 2 : 1 222. The length of the diagonal BD of the parallelogram ABCD is 18 cm. If P and Q are the centroid of the ABC and ADC respectively the length of the line segment PQ is (a) 4 cm (b) 6 cm (c) 9 cm (d) 12 cm 223. If all the sides of a parallelogram are equal and any one angle is 900 then this is a (a) Square (b) Rectangle (d) Trapezium (d) None of these 224. Two tangents QA & QB have been drawn from point Q to a circle, where ∠AQB = 800. Find ∠APB. (a) 500 (b) 600 (c) 700 (d) 480 225. Two equal circles intersect each other at two points. If length of the common chord is 10cm and the distance between centres is 6cm then find the radius of each circle. (a) 5.83 (b) 6.40 (c) 6.84 (d) 7.63 226. BC is diameter of a circle of centre O, If AD bisects ∠BAC then find ∠BCD. (a) 900 (b)600 (c) 450 (d) 480

227. The sum of the interior angles of a polygon is 1440 . The number of sides of the polygon is (a) 6 (b) 9 (c) 10 (d) 12 228. The sum of interior angles of a regular polygon is 1440 . The number of sides of the polygon is (a) 10 (b) 12 (c) 6 (d) 8 229. Among the angles 30 , 36 , 45 , 50 one angle cannot be an exterior angle of a regular polygon. The angle is (a) 30 (b) 36 (c) 45 (d) 50 230. If each interior angle of a regular polygon is 150 , the number of sides of the polygon is (a) 8 (b) 10 (c) 15 (d) ) None of these 231. Each interior angle of a regular polygon is 144 . The number of sides of the polygon is (a) 8 (b) 9 (c) 10 (d) 11 232. If the sum of the interior angles of a regular polygon be 1080 , the number of sides of the polygon is (a) 6 (b) 8 (c) 10 (d) 12 233. An interior angle of a regular polygon is 5 times its exterior angle. Then the number of sides of the polygon is (a) 14 (b) 16 (c) 12 (d) 18 234. If the ratio of an external angle and an internal angle of a regular polygon is 1 : 17, then the number of sides of the regular polygon is (a) 20 (b) 13 (c) 36 (d) 12 235. Ratio of the number of sides of two regular polygons is 5 : 6 and the ratio of their each interior angle is 24 : 25. Then the number of sides of these two polygons is (a) 20, 24 (b) 15, 18 (c) 10, 12 (d) 5, 6 236. In a regular polygon if one of its internal angle is greater than the external angle by 132 , then the number of sides of the polygon is (a) 14 (b) 12 (c) 15 (d) 16 237. If the sum of interior angles of a regular polygon is equal to two times the sum of exterior angles of that polygon, then the number of sides of that polygon is (a) 5 (b) 6 (c) 7 (d) 8 238. There are two regular polygons with number of sides equal to (n – 1) and (n + 2). Their exterior angles differ by 6 . The value of n is (a) 14 (b) 12 (c) 13 (d) 11



239. The ratio between the number of sides of two regular polygons is 1 : 2 and the ratio between their interior angles is 2 : 3. The number of sides of these polygons is respectively (a) 6, 12 (b) 5, 10 (c) 4, 8 (d) 7, 14 240. The sum of all interior angles of a regular polygon is twice the sum of all its exterior angles. The number of sides of the polygon is (a) 10 (b) 8 (c) 12 (d) 6 241. Measure of each interior angle of a regular polygon can never be: (a) 150 (b) 105 (c) 108 (d) 144 242. Each internal angle of regular polygon is two times its external angle. Then the number of sides of the polygon is: (a) 8 (b) 6 (c) 5 (d) 7 243. The number of sides in two regular polygons are in the ratio 5 : 4 and the difference between each interior angle of the polygons is 6 . Then the number of sides is: (a) 15, 12 (b) 5, 4 (c) 10, 8 (d) 20, 16 244. The difference between the exterior and interior angles at a vertex of a regular polygon is 150 . The number of sides of the polygon is (a) 10 (b) 15 (c) 24 (d) 30 245. In a regular polygon, the exterior and interior angles are in the ratio 1: 4 The number of sides of the polygon is (a) 10 (b) 12 (c) 15 (d) 16 246. Each interior angle of a regular polygon is three times is three times its exterior angle, then the number of sides of the regular polygon is: (a) 9 (b) 8 (c) 10 (d) 7 247. A parallel line has been drawn parallel to each side from each vertex of a What will be the ratio of perimeter of new and the old (a) 3 : 2 (b) 4 : 1 (c) 2 : 1 (d) 2 : 3 248. In ABC, the internal & external bisectors of ∠B & ∠C meet at P & Q respectively then ∠PBQ ∠PCQ ? (a) 1800 (b) 1200 (c) 1600 (d) 900 249. In given figure AB ‖ CD then find the value of

(a) 1800 (b) 2700 (c) 3600 (d) 900 250. In the given figure AD ‖ BC & internal bisectors of ∠B and ∠A meet at point O. Find the measure of ∠AOB in degree.

(a) 900 (b) 1050 (c) 1200 (d) 1400



251. Inside a square ABCD, BEC is an equilateral triangle. If CE and BD intersect at O, then ∠BOC is equal to (a) 60 (b) 75 (c) 90 (d) 120 252. ABCD is a parallelogram, its diagonals AC and BD intersect each other at point O. ∠DAC = 320 and ∠AOB = 700 then find ∠DBC. (a) 300 (b) 1020 (c) 380 (d) 480 253. ABCD and ADEF are parallelograms in the given figure. If CA = AF and ∠ACD = 600 then DECF = ?

(a) 300 (b) 600 (c) 750 (d) 800 254. In the figure given below, if x ‖ y ‖ z then find the value of ∠A and ∠B.

(a) 1550. 250 (b) 1600, 200 (c) 1300, 300 (d) None of these 255. In the figure given below, line AB and CD intersect each other at point O. If ∠AOC = 1250, find the value of ∠BOD

(a) 650 (b) 550 (c) 1250 (d) 1400

256. Q is a point in the interior of a rectangle ABCD. If QA = 3 cm, QB = 4 cm and QC = 5 cm, then the length of QD (in cm) is

(a) 3√2 (b) 5√2 (c) √34 (d) √4



257. If the opposite sides of a quadrilateral and also its diagonals are equal, then each of the angles of the quadrilateral is (a) 90 (b) 120 (c) 100 (d) 60 258. ABCD is a rectangle where the ratio of the length of AB and BC is 3 : 2. If P is the mid-point of AB, then the value of sin∠CPB is

(a)

(b)

(c)

(d)

259. In the figure given below ∠POR and ∠QOR from a linear pair. If a – B = 800, then find the value of a and b.

(a) 1400, 400 (b) 1300, 500 (c) 1600, 200 (d) 1200, 600 260. The length of the sides forming the right angle in a right angled triangle is ‘a’ and ‘b’. Three squares are inscribed outwards on the three sides of the triangle. What is the total sum of the area of the triangles and that of the squares so formed? (a) 2(a2 + b2) + ab (b) 2(a2 + b2) + 2.5 ab (c) 2(a2 + b2) + 0.5 ab (D) 2.5(a2 + b2) 261. What is the point of concurrent of altitudes in a triangle called (a) Circumcentre (b) Othocentre (c) Incentre (d) Centroid 262. AC is the diameter of the circumcircle of the cyclic quadrilateral ABCD. If ∠BDC = 420, then the measure of ∠ACB is (a) 420 (b) 450 (c) 480 (d) 580

263. The perimeters of two similar triangles ABC and PQR are 36 cm and 24 cm respectively. If PQ = 10 cm. then AB is (a) 15 cm (b) 12 cm (c) 14 cm (d) 26 cm 264. In ABC. Two points D and E are taken on the lines AB and BC respectively in such a way that AC is parallel to DE. Then ABC and DBE are (a) similar only if D lies outside the line segment AB (b) congruent only if D lies outside the line segment AB (c) always similar (d) always congruent 265. In PQR, S and T are points on sides PR and PQ respectively such that ∠PQR = ∠PST. If PT = 5 cm, PS = 3 cm and TQ = 3 cm, then length of SR is

(a) 5 cm (b) 6 cm (c)

cm (d)

cm

266. In ABC and DEF, AB = DE and BC = EF. Then one can infer that ABC DEF, when (a) ∠BAC = ∠EDF (b) ∠ACB = ∠EDF (c) ∠ACB = ∠DFE (d) ∠ABC = ∠DEF 267. Point O is the in centre of ∠ 3 ∠ (a) 100 (b) 105 (c) 110 (d) 90 268.



In the above figure, O is the cnetre of the circle , OA = 3 cm, AC = 3 cm and OM ⊥ AC then ∠ABC is (a) 600 (b) 450 (c) 300 (d) None of these 269.

In the given figure PQ ‖ RS, then ∠NMS is equal to (a) 200 (b) 230 (c) 270 (d) 470 270. Which angle of the following is equal to two third of its complements? (a) 360 (b) 450 (c) 480 (d) 600 271.

In the given figure AB ‖ CD. If ∠BAF = 980 and ∠AFC = 1440, then the value of ∠ECD is (a) 620 (b) 640 (c) 820 (d) 840

272. In ABC, ∠A = 90 and AD⊥ BC where D lies on BC. If BC = 8 cm, AC = 6 cm, then ABC : ACD = ? (a) 4 : 3 (b) 25 : 16 (c) 16 : 9 (d) 25 : 9 273. In a right-angled triangle ABC, ∠ABC = 90 , AB = 5 cm and BC = 12 cm. The radius of the circum-circle of the triangle ABC is (a) 7.5 cm (b) 6 cm (c) 6.5 cm (d) 7 cm



274. In a right angled ABC ∠ABC = 90 ; BN is perpendicular to AC, AB = 6 cm, AC = 10. Then AN : NC is (a) 3 : 4 (b) 9 : 16 (c) 3 : 16 (d) 1 : 4 275. In a triangle ABC, ∠BAC = 90 and AD is perpendicular to BC. If AD = 6 cm and BD = 4 cm, then the length of BC is (a) 8 cm (b) 10 cm (c) 9 cm (d) 13 cm

276. Suppose ABC be a right-angled triangle where ∠A = 90 and AD⊥ BC. If ABC = 40 cm2, ACD = 10 cm2 and ̅̅̅̅ = 9 cm, then the length of BC is (a) 12 cm (b) 18 cm (c) 4 cm (d) 6 cm

277. In ABC, ∠BAC = 90 and AB =

B. Then the measure of ∠ACB is:

(a) 60 (b) 30 (c) 45 (d) 15

278. For a triangle, base is 6√3 cm and two base angle are 30 and 60 . Then height of the triangle is

(a) 3√3 cm (b) 4.5 cm (c) 4√3 cm (d) 2√3 cm 279. ABC is a right-angled triangle with AB = 6cm and BC = 8 cm. A circle with centre O has been inscribed inside ABC. The radius of the circle is (a) 1 cm (b) 2 cm (c) 3 cm (d) 4 cm

280. BL and CM are medians of ABC rigtht-angled at A and BC = 5 cm, If BL = √

cm, then the length of CM is

(a) 2√5 cm (b) 5√2 cm (c) 10√2 cm (d) 4√5cm 281. If the sides of a right angled triangle are three consecutive integers, then the length of the smallest side is (a) 3 units (b) 2 units (c) 4 units (d) 5 units 282. If each angle of a triangle is less than the sum of the other two, then the triangle is (a) obtuse angled (b) right angled (c) acute angled (d) equilateral 283. If the length of the three sides of a triangle are 6 cm, 8 cm and 10 cm, then the length of the median to its greatest side is (a) 8 cm (b) 6 cm (c) 5 cm (d) 4.8 cm 284. The ortho centre of a right angled triangle lies (a) outside the triangle (b) at the right angular vertex (c) on its hypotenuse (d) within the triangle 285. D and E are two points on the sides AC and BC respectively of ABC such that DE = 18 cm, CE = 5 cm and ∠DEC = 90 . If tan ∠ABC = 3.6, then AC : CD = (a) BC : 2 CE (b) 2 CE : BC (c) 2 BC : CE (d) CE : 2 BC 286. A point D is taken from the side BC of a right-angled triangle ABC, where AB is hypotenuse. Then (a) AB2 + CD2 = BC2 + AD2 (b) CD2 + BD2 = 2 AD2 (c) AB2 + AC2 =2AD2 (d) AB2 = AD2 + BD2 287. In a right-angled triangle, the product of two sides is equal to half of the square of the third side i.e., hypotenuse. One of the acute angle must be (a) 60 (b) 30 (c) 45 (d) 15 288. If the median drawn on the base of a triangle is half its base, the triangle will be: (a) right-angled (b) acute-angled (c) obtuse-angled (d) equilateral 289. ABC is a right angled triangle. Right angled at C and P is the length of the perpendicular from C on AB. If a, b and c are the length of the sides BC, CA and AB respectively, then

(a)

(b)

+

(c)

(d)

290. The length of the three sides of a right angled triangle are (x – 2)cm, x cm and (x+2)cm respectively. Then the value of x is (a) 10 (b) 8 (c) 4 (d) 0 291. Two medians AD and BE of ABC intersect at G at right angles. If AD = 9 cm and BE = 6 cm, then the length of BD (in cm) is (a) 10 (b) 6 (c) 5 (d) 3 292. If the measures of the sides of triangle are (x2 – 1), (x2 + 1) and 2x cm, then the triangle would be (a) equilateral (b) acute-angled (c) isosceles (d) right-angled 293.



In the figure drawn above AB ‖ CD. If ∠DCE = x and ∠ABE = y, then the measure of ∠CEB is] (a) y – x (b) (x + y)/2 (c) x + y – ( /2) (d) x + y – 294.

In the figure drawn above LM ‖ QR. If PQR is divided by LM in such a way that the area of trapezium LMRQ is twice

the area of PLM, then

will be equal to

(a)

√ (b)

√ (c)

(d)

295. Let there be two points A and B. What is the locus of point P, such that ∠APB = 900? (a) The line segment AB itself (b) The point P itself. (c) The circumference of the circle with AB as its diameter. (d) The right bisector of AB. 296. The two medians AD and BE of a triangle ABC intersect each other at point G making right angle. If AD = 9 cm and BE = 6 cm then the length of BD is (a) 10 cm (b) 6 cm (c) 5 cm (d) 3 cm 297. I is the incentre of ABC, ∠ABC = 60 and ∠ACB = 50 . Then ∠BIC is: (a) 55 (b) 125 (c) 70 (d) 65 298. I is the incentre of a triangle ABC. If ∠ABC = 65 and ∠ACB = 55 , then the value of ∠BIC is (a) 130 (b) 120 (c) 140 (d) 110 299. If two angles of a triangle are 21 and 38 , then the triangle is (a) Right-angled triangle (b) Acute-angled triangle (c) Obtuse-angled triangle (d) Isosceles triangle 300. A man goes 24 m due west and then 10 m due north. Then the distance of him from the starting point is (a) 17 m (b) 26 m (c) 28 m (d) 34 m 301. The side of a triangle are in the ratio 3 : 4 : 6. The triangle is: (a) acute-angled (b) right-angled (c) obtuse-angled (d) either acute-angled or right-angled 302. In ABC, ∠C is an obtuse angle. The bisectors of the exterior angles at A and B meet BC and Ac produced at D and E respectively. If AB = AD = BE, then ∠ACB = (a) 105 (b) 108 (c) 110 (d) 135 303. If the length of the sides of a triangle are in the ratio 4 : 5 : 6 is 3 cm, then the altitude of the triangle corresponding to the largest side as base is: (a) 7.5 cm (b) 6 cm (c) 10 cm (d) 8 cm 304. The sum of three altitudes of a triangle is (a) equal to the sum of three sides (b) less than the sum of sides (c) greater than the sum of sides (d) twice the sum of sides



305. In a triangle ABC, the side BC is extended up to D. Such that CD = AC, if ∠BAD = 109 and ∠ACB = 72 then the value of ∠ABC is (a) 35 (b) 60 (c) 40 (d) 45 306. ABC is a triangle. The bisectors of the internal angle ∠B and external angle ∠C intersect at D. If ∠BDC = 50 , then ∠A is (a) 100 (b) 90 (c) 120 (d) 60 307. Taking any three of the line segments out of segments of length 2 cm, 3 cm, 5 cm, and 6 cm, the number of triangles that can be formed is: (a) 3 (b) 2 (c) 1 (d) 4 308. If the circumcentre of a triangle lies outside it, then the triangle is (a) Equilateral (b) Acute angled (c) Right angled (d) Obtuse angled 309. In ABC, AD is the internal bisector of ∠A, meeting the side BC at D. If BD = 5 cm, BC = 7.5 cm, then AB : AC is (a) 2 : 1 (b) 1 : 2 (c) 4 : 5 (d) 3 : 5 310. O and C are respectively the orthocentre and circumcentre of an acute-angled triangle PQR. The Points P and O are joined and produced to meet the side QR at S. If ∠PQS = 60 and ∠QCR = 130 , then ∠ RPS = (a) 30 (b) 35 (c) 100 (d) 60 311. In parallelogram ABCD, the length of the diagonal BD is 18 cm. If points P and Q are the centroid of and respectively, then the length of the line segment PQ is (a) 4 cm (b) 6 cm (c) 9 cm (d) 12 cm 312. The radius of a circle with O as its centre is 4 cm. PR is a tangent drawn at point Q on the circle. If ∠POR = 900,

OR = 5 cm and OP =

cm then the length of PR is

(a) 3 cm (b)

(c)

cm (d)

cm

313. The radii of two concentric circles are 9 cm and 15 cm. If a chord of the bigger circle is tangent on the smaller circle then the length of that chord is (a) 24 cm (b) 12 cm (c) 30 cm (d) 18 cm 314. In ABC, point P and Q are the mid points of the side AB and AC respectively. R is point on PQ and PR : RQ = 1 : 2. If PR = 2 cm, then measure of BC is (a) 4 cm (b) 2 cm (c) 12 cm (d) 6 cm 315. If ABC is an isosceles triangle with ∠C = 90 and AC = 5 cm, then AB is:

(a) 5 cm (b) 10 cm (c) 5√2 cm (d) 2.5 cm 316. In a triangle ABC, AB = AC, ∠BAC = 40 Then the external angle at B is: (a) 90 (b) 70 (c) 110 (d) 80 317. ABC is an isosceles triangle such that AB = AC and ∠B =35 . AD is the median to the base BC. The ∠BAD is: (a)70 (b)35 (c)110 (d) 55 318. If FGH is isosceles and FG < 3 cm, GH = 8 cm, then of the following, the true relation is. (a) GH = FH (b) GF = GH (c) FH > GH (d) GH < GF 319. ABC is an isosceles triangle such that AB = AC and AD is the median to the base BC with ∠ABC = 35 . Then ∠BAD is (a) 35 (b) 55 (c) 70 (d) 110 320. If angle bisector of a triangle bisect the opposite side, then what type of triangle is it? (a) Right angled (b) Scalene (c) Similar (d) Isosceles

321. ABC is an isosceles triangle and ̅̅ ̅̅ ̅̅ ̅̅ = 2a unit, ̅̅ ̅̅ = a unit. Draw ̅̅ ̅̅ ⊥ ̅̅ ̅̅ , and find the length of ̅̅ ̅̅ .

(a) √ 5 a unit (b) √

a unit (c) √ (d)

√

a unit

322. ABC is an isosceles triangle with AB = AC. The side BA is produced to D such that AB = AD. If ∠ABC = 30 , then ∠BCD is equal to (a) 45 (b) 90 (c) 30 (d) 60 323. An isosceles triangle ABC is right – angled at B.D is a point inside the triangle ABC. P and Q are the feet of the perpendiculars drawn from D on the side AB and AC respectively of ABC. If AP = a cm, AQ = b cm and ∠BAD = 15 , sin 75 =



(a)

√ (b)

(c)

√

(d)

√

324. In an isosceles triangle, if the unequal angle is twice the sum of the equal angles, then each equal angle is (a) 120 (b) 60 (c) 30 (d) 90 325. ABC is an isosceles triangle with AB = AC. A circle through B touching AC at the middle point intersects AB at P. Then AP : AB is: (a) 4 : 1 (b) 2 : 3 (c) 3 : 5 (d) 1 : 4 326. A, B, C are three points on a circle. Tangent drawn at point A intersects BC (when produced) at point T. ∠BTA = 400, ∠CAT = 440. Then the measure of the angle subtended by BC at the centre of the circle is (a) 840 (b) 920 (c) 960 (d) 520 327. Which one of the following values can never be the measure of an internal angle of a regular polygon? (a) 1500 (b) 1050 (c) 1080 (d) 1440 328. The two chords AB and AC are 8 cm and 6 cm long respectively. If ∠ BAC = 90 then radius of the circle is (a) 25 cm (b) 20 (c) 4 cm (d) 5 cm

329. In 2 3 (a) 6cm (b) 9 cm (c) 12 cm (d) 8 cm 330. G is the centroid of the equilateral ABC. If AB = 10 cm then length of AG is

(a) √

cm (b)

√

cm (c) 5√3 cm (d) 10√3 cm

331. If the circum radius of an equilateral triangle be 10 cm, then the measure of its in - radius is (a) 5 cm. (b) 10 cm. (c) 20 cm. (d) 15 cm. 332. If the three medians of a triangle are same then the triangle is (a) equilateral (b) isosceles (c) right – angled (d) obtuse - angle 333. The radius of the incircle of the equilateral triangle having each side 6 cm is

(a) 2√3 cm (b) √3 cm (c) 6√3 cm (d) 2 cm 334. ABC is an equilateral triangle and CD is the internal bisector of ∠C. If DC is produced to E such that AC = CE, then ∠ CAE is equal to (a) 45 (b) 75 (c) 30 (d) 15 335. Let ABC be an equilateral triangle and AX, BY, CZ be the altitudes. Then the right statement out of the four given responses is (a) AX = BY = CZ (b) AX BY = CZ (c) AX = BY CZ (d) AX BY CZ 336. If ABC is an equilateral triangle and P, Q, R respectively denote the middle points of AB, BC, CA then. (a) PQR must be an equilateral triangle (b) PQ + QR + PR = AB (c) PQ + QR + PR = 2 AB (d) PQR must be a right angled triangle 337. In a triangle, if orthocentre, circumcentre, incentre and centroid coincide, then the triangle must be (a) obtuse angled (b) isosceles (c) equilateral (d) right – angled 338. If the in centre of an equilateral triangle life inside the triangle and its radius is 3 cm, then the side of the equilateral triangle is

(a) 9√3 cm (b) 6√3 cm (c) 3√3 cm (d) 6 cm 339. The side QR of an equilateral triangle PQR is produced to the point S in such a way that QR = RS and P is joined to S. Then the measure of PSR is (a) 30 (b) 15 (c) 60 (d) 45 340. If ABC is an equilateral triangle and D is a point on BC such that AD BC, then

(a) AB : BD = 1 : 1 (b) AB : BD = 1 : 2 (c) AB : BD = 2 : 1 (d) AB : BD = 3 : 2 341. The in-radius of an equilateral triangle is of length 3 cm. The length of each of its medians is

(a) 12 cm (b)

cm (c) 4 cm (d) 9 cm

342. In a triangle, if three altitudes are equal, then the triangle is (a) Obtuse (b) Equilateral (c) Right (d) Isosceles 343. If in a triangle, the circumcentre, in centre, centroid and ortho centre coincide, then the triangle is (a) Acute angled (b) Isosceles (c) Right angled (d) Equilateral



344. If the orthocenter and the centroid of a triangle are the same, then the triangle is: (a) Scalene (b) Right angled (c) Equilateral (d) Obtuse angled 345. The tangents drawn at point A and B on a circle with centre O, intersect at point P. If in quadrilateral PAOB, ∠AOB : ∠APB = 5 : 1. Then measure of ∠APB is (a) 300 (b) 600 (c) 450 (d) 150 346. D is a point on side AC of ABC, if points P, Q, X and Y are mid points of AB, BC, Ad and DC respectively. Then PQ : XY = (a) 1 : 2 (b) 1 : 1 (c) 2 : 1 (c) 2 : 3 347. AB is a chord of a circle at PAT is tangent at point A and C is point on the circle . If ∠BAT = 750 and ∠BAC = 450, then measure of ∠BAC is (a) 400 (b) 450 (c) 300 (d) 700 348. Point O is the incentre of ⊥ ∠ 5 then the measure of ∠ABC is (a) 75 (b) 45 (c) 150 (d) 90 349. The radius of the incircle in an equilateral triangle is 3cm. Then the length of its each median is

(a) 12 cm (b)

(c) 4 cm (d) 9 cm

350. Tangents are drawn at the end points of AB which is a diameter of a circle with centre P. If a tangent at point C intersects the other tangents drawn through points A and B at points Q and R then the measure of ∠QPR is (a) 45 (b) 60 (c) 90 (d) 180 351. Two circles touch at point A externally and PQ is a common tangent drawn at these two circle touching them at point P and Q. Then the value of ∠ PAQ is (a) 45 (b) 90 (c) 80 (d) 100

352. In two points D and E are on the sides AB and AC respectively such that AD =

AC. If the length of BC is 15

cm then the length of DE is (a) 10 cm (b) 8 cm (c) 6 cm (d) 5 cm 353. AB = 8 cm and CD = 6cm are two such parallel chords that lie in the same side of the centre of a circle. The distance between these two chords is 1 cm. Then the radius of the circle is (a) 5 cm (b) 4 cm (c) 3 cm (d) 2 cm 354. In ∠ 5 5 (a) 2 : 1 (b) 1 : 2 (c) 4 : 5 (d) 3 : 5 355. The length of a chord of a circle is equal to its radius. The value of the angle subtended by this chord at the centre of the circle is (a) 30 (b) 45 (c) 60 (d) 90 356. Point O and C are the orthocenter and the circum centre of an acute angle triangle PQR respectively . When points P and Q are joined and produced it intersects side QR at point S. If ∠PQS = 60 and ∠QCR =130°,then ∠ RPS is (a) 30 (b) 35 (c) 100 (d) 60 357. Point O is the centroid of 5 area fo quadrilateral BDOF is (a) 20 cm2 (b) 30 cm2 (c) 40 cm2 (d) 25 cm2 358. In the the medians CD and BE intersect each other at point O then the ratio of the area of ODE and will be (a) 1 : 6 (b) 6 : 1 (c) 1 : 12 (d) 12 : 1 359. If the radius of the circum circle of an equilateral triangle is 10 cm, then what will be the radius of the incircle inscribed in it? (a) 5 cm (b) 10 cm (c) 20 cm (d) 15 cm 360. What will the distance between two parallel chords each measuring 8 cm in a circle, whose diameter is 10 cm long? (a) 6 cm (b) 7 cm (c) 8 cm (d) 5.5 cm 361. ABCD is a rhombus. A line passing through point C intersects AD and AB at points P and Q respectively.

When AB and AD are produced, If DP =

AB then BQ : AB is



(a) 2 : 1 (b) 1 : 2 (c) 1 : 1 (d) 3 : 1 362. A line drawn parallel to BC in intersects its side AB and AC at point D and E respectively. If area of ABE is 36 cm2then the area of is (a) 18 cm2 (b) 36 cm2 (c) 18cm (d) 36 cm 363. Two chords AB and CD of a circle with centre O, meet at point P. If ∠AOC 50 ∠ 4 ∠ (a) 60 (b) 40 (c) 45 (d) 75 364. ABCD is a square Point M and N are mid points of the side AB and BC respectively. DM and AN intersect at point O. Then which of the following is correct? (a) OA : OM = 1 : 2 (b) AN = MD (c) ∠ADM = ∠ANB (d) ∠AMD = ∠BAN

365. If

the value of P +

(a) 4 (b) 5 (c) 10 (d) 12 366. x, x + 1 and x – 1 are the sides of a right angled then its hypotenuse will be: (a) 5 (b) 4 (c) 1 (d) 0 367. If the sum of the interior angles of a polygon is 10800, find the number of its sides (a) 8 (b) 6 (c) 10 (d) 9 368. The sum of the interior angles of a polygon is 1600 the number of sides of the polygon is (a) 15 (b) 18 (c) 20 (d) 30 369. If the sum of the internal angles of a polygon is 1560 then the number of the sides of the polygon is (a) 8 (b) 10 (c) 12 (d) 15 370. In adjoining figure ∠DEC = 700, BC ‖ DE and ∠CBA = 840. If ∠BAC = x0 then find the value of x.

(A) 300 (b) 260 (c) 180 (d) 280 371. In the figure, CE ‖ BD and ∠BAD = 1100, ∠ABD = 300, ∠ADC = 75 ∠ = 600, then what will be the value of x?

(a) 450 (b) 750 (c) 850 (d) 1200 372. In the adjoining figure ∠PQA = 200 and ∠APQ = 1200 then find the value of ∠PAQ.



(a) 1200 (b) 400 (c) 200 (d) 600 373. In the given figure ∠A = 600 and ∠ABC = 800, then ∠BPC is

(a) 400 (b) 450 (c) 200 (d) 300 374. If each interior angle of a regular polygon is 1350 then find the number of sides (a) 6 (b) 8 (c) 5 (d) 9 375. OA and OB are radii of a circle with centre O ∠AOB 2 0. Tangents drawn at points A and B meet at point C. If OC divides the circle into two equal parts at point D then point D divides the side OC in the ratio of (a) 1 : 2 (b) 1 : 3 (c) 1 : 1 (d) 2 : 3 376. In the figure AB ‖ CD, ∠BAE = 1050, ∠AEC = 250 then ∠DCE is



(a) 1300 (b) 800 (c) 1550 (d) 750 377. In the given ROQ is a diameter. If ∠POR = 1300 then ∠QPO is

(a) 400 (b) 450 (c) 500 (d) 750 378. In adjoining figure ∠CAB = 900 and AD ⊥ BC. If AC = 100 cm, AB = 100 cm and BC = 125 cm, then the length of CD is

(a) 50 cm (b) 37.5 cm (c) 62.5 cm (d) 65 cm 379. In the adjoining figure ∠AOB = 900 then ∠APB is



(A) 300 (b) 450 (c) 250 (d) data insufficient 380. In the adjoining figure ∠BAT = 650, then find ∠BDA.

(a) 650 (b) 1150 (c) 51300 (d) 850 381. In acute angled ∠ 600. If O is the orthocentre of ∠ ∠ (a) 1500 (b) 1200 (c) 600 (d) 300 382. In the adjoining figure, ∠OAB = 200, ∠OCB = 300 then the value of ∠AOC is

(a) 800 (b) 1000 (c) 500 (d) 600 383. AB and CD are the diameters of circle C (O, r). if ∠OBD = 500 then find the value of ∠AOC. (a) 800 (b) 400 (c) 1000 (d) 250 384. In the adjoining figure ∠ABC = 450, then ∠CDT is



(a) 150 (b) 200 (c) 250 (c) 300 385. In the given figure AB ‖ CD, ∠EFC = 300 and ∠ECF = 100 ∠

(a) 1300 (b) 700 (c) 1000 (d) 800 386. The areas of two similar triangles are 96 cm2 and 150 cm2. If the largest side of the larger triangle is 20 cm the largest side of the smaller triangle is (a) 15 cm (b) 16 cm (c) 18 cm (d) 20 cm 387. The area of a rhombus is 120cm2. If the length of one of its diagonal is 10 cm, the length of its one side is

(a) 12 cm (b) 13 cm (c) 24 cm (d) 2√3 cm 388. In the given figure ∠ABC = 100 ∠ 2 ‖ ∠

(a) 80 (b) 600 (c) 400 (d) 200 389. In the adjoining AB is the diameter of a circle. ∠BOD = 150 and ∠EOA = 850, then ∠ECA is



(a) 200 (b) 350 (c) 400 (d) None of these 390. In ∠ 6 is the mid point of AC. Find the length of BC.

(A) 4 cm (b) √6 (c) 3 cm (d) 3.5 cm 391. In the given figure ∠ADC = 140 and AOB is the diameter of the circle then ∠BAC is

(a) 400 (b) 500 (c) 700 (d) 750 392. In the given figure ∠QPR = 67 ∠ 2 is the diameter of the circle, then ∠QRS is



(a) 410 (b) 230 (c) 670 (d) 180 393. In the given figure PQRS is a rectangle, whose area is 8 cm 4 cm. Triangle are equilateral and the radii of each circle is 1 cm. Then the perimeter of ABCDEFGHIJKLMNA is

(a) 47.84 cm (b) 38.84 cm (c) 36.84 cm (d) 34.84 cm 394. The vertex A of a vertical pillar is at the ground. C is the midpoint of AB. BC subtends an angle tan

(a)

(b)

(c)

(d)

395. A, B, C are points such that AB = 10 cm and BC = 6 cm. Then AC is (a) 2 cm (b) 3 cm (c) 5 cm (d) 16 cm 396. ABCD is a parallelogram. If E and F are two points on sides DC and AD respectively. If areas of respectively, then



(a) (b) 2 (c) 2 (d) 2 3 397. If PL, QM and RN are altitudes of O is the orthocentre of PQR, then Q will be the ortho centre of (a) (b) PRn (c) (d) 398. In an isosceles ABC, a perpendicular is drawn through the point P on BC, it intersect AB at Q, and meet CA at R when CA is produced to R. then (a) AQR is equilateral (b) AQR is isosceles (c) Q is the mid point of AB (c) A is the midpoint of CR 399. The ratio of the internal and the external angle of a regular polygon is 2 : 1. The number of the sides of the polygon is (a) 3 (b) 5 (c) 6 (d) 12 400. In the following figure the area of the equilateral triangle inscribed in the square of side ‘a’ will be

(a) √

a2 (b)

√

(c)

√

(d)

√

401. The interior bisector of ∠B and ∠C of ∠ then the value of ∠BOC is (a) 500 (b) 1000 (c) 1300 (d) 1600 402. In ABC the medians BE and CF intersect at point G. If the straight line AGD, intersect BC at point D such that GD = 1.5 cm then the length of AD is (a) 2.5 cm (b) 3 cm (c) 4 cm (d) 4.5 cm 403. Three lines OA OB and OC are drawn through point O If OP and OQ are the bisector of ∠BOA and ∠AOC respectively and ∠POQ 0, then a ∠ BOC is an obtuse angle b ∠BOC is an acute angle c BOC is a straight line d ∠BOC is a right angle 404. In the adjacent figure, O is the centre of the circle. If tangent PQ = 12 lcm and BQ = 8 cm. Then the length of chord AB is



(a) 10 cm (b) 4√5 cm (c) 4 cm (d) 18 cm 405. The radii of two non intersecting circles are R and r and one circle is inscribed inside the other circle. If the least distance between their circumference is S then the distance (a) R – r + s (b) R – r – s (C) R + r – s (d) R – r 406. In intersect at O and BE + EO = Bo, then (a) CO + OF = CF (b) CF + FO = CO (c) FC CO = FO (d) CF + FO = CO 407. If midpoint of the sides BC, CA and AB of and BE intersect at point G inside the

(a) CG + GF > CF (b) CG + GF < CF (c) CG + GF = CF (d) CG = GF =

408. The radii of two circles are 15 cm and 20 cm respectively. Their centres are at a distance of 25 cm. Find the common length of the common chord. (a) 24 cm (b) 25 cm (c) 15 cm (d) 20 cm 409. A line XY drawn parallel to the BC the base of meet AB and AC at point X and Y respectively. If AB = 5 BX and YC = 3 cm, then AY is (a) 8 cm (b) 15 cm (c) 10 cm (d) 12 cm 410. In triangle ABC, D, E, F are the mid points of sides AB, BC and CA respectively. If the area of then the area of DEF is

(a)

(b)

(c)

(d)

411. What will be the length of the chord that subtends an angle of 900 at the cnetre of a circle with the radius of unit length?

(a) √2 (b) √3 (c)

√ (d)

√

412. A circle C passes through three non-collinear points D, E and F. Where DE = EF = DF = 3 cm. Then the radius of the circle is

(a) √

(b) √3 (c)

√ cm (d)

√

413. In circle, a square and an equilateral triangle have been inscribed. If their sides are a and b, respectively. Then

(a) a2 =

(b)

(c) 3b2 = 2a2 (d) 3a2 = 2b2

414. In ∠ If AD ⊥ BC then BC AD is (a) BD.AC (b) AB.CD (c) AB.AC (d) AC.BC 415. In the given figure ∠ABC = 90 6



(a) 3 cm (b) 4 cm (c) 5 cm (d) 6 cm 416. The length of a chord is equal to the length of the radius. Then the angle subtended by the chord in the major segment of the circle. (a) 300 (b) 450 (c) 600 (d) 900 417. In the parallelogram ABCD, the diagonals bisect each other at point O. is an equilateral triangle with each side 6 cm long. The length of the diagonal AC is

(a) 3√3 cm (b) 6√3 cm (c) 3√6 cm (d) 12 cm 418. In cyclic quadrilateral ABCD, the sides AB and CD meet at point P when produced and sides AD and BC meet at point Q when produced. If ADC = 850 and ∠BPC 4 0 then ∠CQD is (a) 300 (b) 450 (c) 600 (d) 750 419. In Then which of the following statement is true? (a) MN = YZ (b) NY = NZ = MN (c) MX = MY = NY (d) MN = MX = MY 420. If AD, BE and CF be the medians of statements is true?

(a) (AD + BE + CF) = (AB + BC + CA) (b) (AD + BE + CF) >

(c) (AD + BE + CF) <

(AB + BC + CA) (d) (AD + BE + CF) =

421. ABCD is a trapezium in which AB ‖ DC If the diagonals intersect each other at point O then which of the following is true?

(a)

(b)

(c)

(d)

422. A rectangle ABCD is inscribed in a circle with centre O If AC be the diagonal and ∠BAC 3 0 then the radius of the circle is equal to

(a) √

(b) BC (c) √3 BC (d) 2 BC

423. In the adjoining figure AB ‖ CD ‖ EF then ∠CEF will be equal to

(a) 1200 (b) 1350 (c) 1500 (d) 1600 424. Let P be the set of squares, Q the set of parallelograms R the set of quadrilaterals and S, the set of rectangles Then which of the following is true? 1. P 2 R P 3 P S 4 S R (a) 1, 2, 3 (b) 1, 3, 4 (c) 1, 2, 4 (d) 3, 4 425. ABCD is a rhombus and K is the midpoint of AB CK ⊥ AB Find the value of ∠A (a) 1200 (b) 1050 (c) 900 (d) 600 426. The centres of the two circular wheels with same radius ‘r’ is at a distance of a The least length of the belt



(a) 2(a + (b)

(c) 2a + (d)

427. If D, E and F are the mid point of the side BC, AC and AB of an equilateral triangle ABC. Then the ratio of the areas of

(a) 1.1 : 1 (b) 1 : 1.1 (c) 0.9 : 1 (d) 1 : 1 428. ABCD is a cyclic quadrilateral tangents drawn at A and C meet at P If ∠ABC 0 then ∠APC will be (a) 200 (b) 400 (c) 600 (d) 800 429. A parallelogram ABCD and a rectangle ABEF are drawn between parallel lines EF and CD. If AB = 7cm and BE = 6.5 cm, then area of parallelogram will be (a) 22.75cm2 (b) 11.375cm2 (c) 45.5 cm2 (d) 45.0 cm2 430. In BE intersect at G, then AG + BG + CG is equal to

(a) AD = BE = CF (b)

(AD + BE + CF) (c)

(d)

431. In ∠ ⊥

(a) p2 = b2 + c2 (b)

(c)

(d) p2 = b2c2

432. The area of 6 2. XY is drawn parallel to BC which divides AB in the ratio 3 : 5. If BY is joined then area of (a) 3.5 cm2 (b) 3.7 cm2 (c) 3.75 cm2 (d) 4.0 cm2 433. D is a point on the side AB of ∠ADE ∠ACB then the value of AD.AB is (a) AE. BC (b) AC.DE (c) AE.AC (d) AB.BC 434. In the adjoining figure AB ‖ CD and EF transverse them at point M and N The bisector of ∠M and ∠N meet at point Q If ∠AME 0, find the value of ∠MQN



(A) 600 (b) 700 (c) 800 (d) 900 435. In a plane there are three concurrent lines OA OB and OC and their point of concurrence is O If ∠AOB ∠BOC and ∠COA measure 2x0, 5x0 and 8x0 respectively then the value of x is (a) 240 (b) 180 (c) 150 (d) 120 436. In the adjoining figure AB is the diameter C and D are points on the circle If ∠CAD 3 0 ∠CBA 0, then find the value of ∠ACD

(a) 400 (b) 500 (c) 350 (d) 900 437. In ∠A meets with BC at D If AB 4 AC 3 and ∠A 6 0, then the length of AD will be

(a) 2√3 (b) √

(c)

√

(d)

√

438. In the adjoining figure, a rectangle is inscribed in a circle with centre O. AB > BC The ratio of the area of

circle to that of the rectangle is √3 Line segment DE intersect AB at point E where ∠ODC ∠ADE Then AE AD = ?



(a) 1 : √3 (b) 1 : √2 (c) 1 : 2√3 (d) 1 : 2 439. In the adjoining figure ABCD is a rectangle in which AE = EF = FB. Then the ratio of the areas of

(a) 1 : 4 (b) 1 : 6 (c) 2 : 5 (d) 2 : 3 440. In adjoining figure in ∠ 4 BP and CP are the bisector of ∠B and ∠C respectively. Then ∠BPC is

(a) 900 (b) 1100 (c) 1200 (d) 1400 441. In the adjoining figure if AB BC CD DE EF FG GA then the approximate value of ∠DAE is



(a) 150 (b) 200 (c) 300 (d) 250 442. In the adjoining figure AB is the diameter and the length of the radius is 6.5 cm. If the length of the chord CA is 5 cm, find the area of

(a) 60 cm2 (b) 30 cm2 (c) 40 cm2 (d) 52 cm2 443. In the adjoining figure ABCD is a square and BCE is an equilateral triangle Then the value of ∠DEC is

(a) 150 (b) 300 (c) 200 (d) 450 444. In the adjoining figure OG = OF, then the area of the figure will be

(a) 64m2 (b) 62m2 (c) 60m2 (d) 58m2 445. In quadrilateral ABCD, the line segment DE and CE are the bisectors of ∠C and ∠D respectively then the right answer is



(a) ∠A + ∠B = ∠CED (b) ∠A + ∠B = 2∠CED c ∠A ∠B 3∠CED (d) None of these 446. ABCD is a parallelogram; There is a point P on side AB. If DP and CP are joined in such a way that they bisect ∠ADC and ∠BCD respectively. Then DC is equal (a) CB (b) 2 CB (c) 3 CB (d) 4 CB

447. In the given figure AD = BD = AC; ∠CAE = 75 ∠

(a) 450 (b) 500 (c) 600 (d) 37

448. in the given figure EC ‖ BA, ∠ECD = 70 ∠ 2 ∠

(a) 20 (b) 500 (c) 600 (d) 700 449. If in a triangle ABC, a line is drawn parallel to BC from a point D on AB which intersects AC at E. Then which of the following statements is not true?

(a) AD : DE = AB : BC (b)

(c) ar ( 2 (d)

450. If AD : CB = 2 : 3 and OA : OC = 4 : 7 then find OD : OB



(a) 4 : 7 (b) 12 : 14 (c) 7 : 4 (d) 14 : 12 451. Let ABCD be a square. M, N, R are points on AB, BC and CD respectively, such that AM = BN = CR. If ∠MNR = 900 then ∠MRN is (a) 300 (b) 450 (c) 600 (d) 750 452. In the adjoining figure PR ‖ AB, PQ ‖ BC and QR ‖ CA. The find the relation between AC and QR.

(a) AC =

QR (b) 2 AC2 = QR2 (c) AC = BQ.QR (d) AC.QR = 1

453. ABC is a triangle. A point P is on AB such that ∠ACP = ∠ABC. If AC = 9 cm, CP = 12 cm and BC = 15 cm, then AP is (a) 11.2 cm (b) 10.2 cm (c) 8.0 cm (d) 7.2 cm 454. In the adjoining figure, in = 3 AP, CQ = 3 AQ and BC = 36. Find the value of PQ.

(a) 6 (b) 8 (c) 9 (d) 10 455. Let a triangle . In as OB and OC are the bisector of ∠B and ∠C respectively. If ∠BAC = 600 then ∠BOC is (a) 1200 (b) 1450 (c) 1500 (d) 1550 456. ABCD is a horizontal square. The diagonals of the square intersect at O. A 40 cm long rod OP is vertically situated at O. If the side of the square is 20 cm, then length of side PA is (a) 28.3 cm (b) 35.6 cm (c) 42.3 cm (d) 44.4 cm 457. In the adjoining figure the value of x is



(a) 300 (b) 450 (c) 600 (d) None of these 458. In the adjoining figure, if AP = 6 cm, AB = 2 cm, PC = 8 cm then the length of CD

(a) 4 cm (b) 3 cm (c) 5 cm (d) 6 cm 459. In a circle with centre C, PQ and RS are two parallel chords such that PQ = 8 cm and RS = 16 cm. If the chords are in the same side of the centre and distance between them is 4 cm. Find the radius of the circle.

(a) 3√2 cm (b) 3√5 cm (c) 4√5 cm (d) 5√5 cm 460. In an equilateral triangle is drawn in a circle, then the ratio of a side fo the triangle and the diameter of the circle is

(a) √2 2 (b) √3 2 (c) 1 : √3 (d) 2 : 3 461. In the adjoining figure If PQ = 13 cm, AB = 6 cm, BR = 8.2 cm and PR = 5.2 cm then the lengths of QR and AR are

(a) 8.2 cm and 10.4 cm (b) 4.1 cm and 6 cm (c) 2.6 cm and 5.2 cm (d) 4.1 cm and 10.4 cm 462. In the given figure O is the centre of the circle ∠BAC = 52 ∠



(A) 520 (b) 1040 (c) 1280 (d) 760 463. In the adjoining O is the centre of the circle. If PA = 12 cm, PC = 15 cm and CD = 7 cm, then find the length of AB.

(a) 5 cm (b) 10 cm (d) 2 cm (d) 9 cm 464. In ⊥ ∠ ∠

(a) 600 (b) 700 (c) 800 (d) None of these 465. A secant drawn from an external point O, intersects the circle at A and B such that OA = 4 cm and OB = 9 cm. Find the length of the tangent drawn from this point on the circle.

(a) √ 3 (b) √5 cm (c) 6 cm (d) √ 466. In the given figure AB ‖ CD, ∠ABO = 40 ∠ 3 ∠



(a) 35 (b) 1100 (c) 700 (d) 1400 467. In the adjoining figure find the value of ∠ABC.

(A) 800 (b) 600 (c) 200 (d) 400 468. In the adjoining figure AD is the bisector of ∠BAC then AB is

(a) 6 cm (b) 5 cm (c) 5.25 cm (d) 5.75 cm 469. If A and B are fixed points and a point P moves in such a way that ∠APB is a right angle then the locus of the point P is (a) None of these (b) A circle (c) An ellipse (d) SA hyperbolic circle 470. In an equilateral ⊥ (a) 2 AB2 = 3 AD2 (b) 3 AB2 = 4 AD2 (c) 5 AB2 = 6 AD2 (d) 4 AB2 = 5 AD2 471. In the adjoining figure two chords AB and CD intersect at point P. If AB = 5 cm and PD = 4 cm, then what will be the length of CD?



(a) 4 cm (b) 3 cm (c) 2.5 cm (d) 2 cm 472. In the given figure AB ‖ CD, ∠ALC = 60 and EC is the bisector of ∠LCD. If EF ‖ AB then the value of ∠CEF is

(A) 1200 (b) 1400 (c) 1500 (d) None of these 473. Two chords AB and CD of a circle intersect each other at point E such that AE = 2.4 cm, BE = 3.2 cm and CE = 1.6 cm. Then length of DE is (a) 4.8 cm (b) 6.4 cm (d) 1.6 cm (d) 3.2 cm 474. Two equal circles with radius r intersect in such a way that they pass through the centre of each other. Find the length of the common chord.

(a) r√3 (b)

√3 (c) √ (d) r√2

475. In the adjoining figure, ABC is an equilateral ∠ ∠

(a) 400, 1400 (b) 300, 1500 (c) 900, 900 (d) 600, 1200 476. A secant drawn from an external point O, intersect the circle at A and B, such that OA = 4cm, OB = 9 cm. Then what will be the length of the tangent drawn from this point on the circle?

(a) √ 3 (b) √5 (c) 6 cm (d) √ 477. In the adjoining figure find the value of ∠QSR



(a) 500 (b) 650 (c) 700 (d) 750 478. In the adjoining figure in ∠ 0 and ∠A = 2x – 200, then the value of ∠B is

(a) 300 (b) 400 (c) 440 (d) 640 479. In the adjoining figure PQ is the tangent and QOR is the diameter of a circle. If ∠QPO = 350 then value of ∠POR is

(a) 1250 (b) 1200 (c) 700 (d) 1150 480. In the adjoining figure AB is the diameter and C and D are points on the circle. If ∠CAD = 300 and ∠CBA = 700 then what will be the value of ∠ACD?

(a) 400 (b) 500 (c) 350 (d) 900 481. In the adjoining figure A, B, C and D are four points on the circle. If AB = 24, BC = 12 then what will be the ratio of the areas of



(a) 1 : 4 (b) 1 : 2 (c) 1 : 3 (d) data insufficient 482. In the adjoining figure the value of x is

(a) 6 cm (b) 7 cm (c) 6.7 cm (d) 7.7 cm 483. In the given ABCD is a cyclic quadrilateral and O is the centre of the circle. If ∠BOC = 1360, then what will be the value of ∠BDC

(a) 1100 (b) 1120 (c)1090 (d) None of these 484. In the given figure o is the centre of the circle. ∠ 25 ∠



(a) 25 (b) 30 (c) 65 (d) 150 485. In the given figure PQ = 12 cm, BQ = 8 cm then the length of the chord is

(a) 10 cm (b) 4√5 cm (c) 4 cm (d) 18 cm 486. In the given figure O is the centre of the circle. If ∠AOC = 1400 then the value of ∠ABC is

(a) 1100 (b) 1200 (c) 1150 (d) 1300 487. What will be the value of x in the given figure?



(a) 400 (b) 250 (c) 300 (d) 450 488. In the given figure O is the centre of the circle. If AB = 16 cm, CP = 6 cm, PD = 8 cm and AP > PB then value of AP is

(A) 12 cm (b) 24 cm (c) 8 cm (d) 6 cm 489. In the given figure O is the centre of the circle then value of x is

(a) 600 (b) 450 (c) 150 (d) 300 490. If O is the centre of the circle, then value of x is



(a) 350 (b) 300 (c) 390 (d) 400 491. In the given figure ∠ADB is

(a) 1320 (b) 1440 (c) 480 (d) 960 492. In the given figure the value of x is

(a) 13 cm (b) 12 cm (d) 16 cm (d) 15 cm 493. In the given figure find the value of x



(a) 16 cm (b) 9 cm (d) 12 cm (d) 7 cm 494. In the given the value of x is

(a) 2.2 cm (b) 1.6 cm (c) 3 cm (d) 2.6 cm 495. A polygon has 27 diagonals. Find the number of its sides (a) 9 (b) 10 (d) 11 (d) 12 496. In the given figure BT and CT are two tangents then ∠A is

(a) 800 (b) 600 (c) 500 (d) 400 497. If ABCD is a cyclic quadrilateral then value of x is



(a) 1100 (b) 800 (c) 700 (d) 1000 498. In the given figure, the value of x is

(a) 600 (b) 900 (c) 700 (d) 400 499. In the given figure O is the centre of the circle then ∠ACB is

(a) 600 (b) 1200 (c) 750 (d) 900 500. In the given AD, AE and BC are tangents Then



(a) AD = AB + BC + CA (b) 2AD = AB + BC + CA (c) 3 AD = AB + BC + CA (d) 4 AD = AB + BC + CA 501. If the chord of a circle is equal to its radius the angle subtended by the chord at the centre of the circle is

(a) 600 (b) 450 (c) 300 (d) 750 502. AB is the diameter of the circle and O is its centre. CD and AB intersect in such a way that OE = EB and CE = 6 cm, ED = 2 cm. Find the radius of the circle.

(a) 4 cm (b) 6 cm (c) 4√3 cm (d) 8 cm 503. AB is the diameter and AC is a chord of a circle and ∠BAC = 300 then which of the statements is correct?



(a) BC > BD (b) BC > BD (c) BC = BD (d) Cannot say 504. In the given figure A, B, C are points on the circumference of the circle and O is the centre. If ∠ABC is

(a) 600 (b) 750 (c) 900 (d) None of these 505. In the given figure AD : DC = 2 : 3 then ∠ABC is

(a) 30 (b) 40 (c) 45 (d) 110 506. In the given figure PQ is a tangent at point K, and LN is the diameter. If ∠KLN = 30 ∠

(a) 300 (b) 500 (c) 600 (d) 700 507. If the angles of a pentagon are in the ratio 1 : 2 : 3 : 5 : 9, then what is the value of the least smallest) angle?



(a) 720 (b) 450 (c) 540 (d) 270 508. ABCD is a parallelogram and E is the midpoint of BC then

(a) AF =

AB (b) AF = 2 AB (c) AF = 3 AB (d) AF2 = 2 AB2

509. The ratio of the corresponding sides of two similar triangles is 1 : 3. What will be ratio of their corresponding altitudes? (a) 1 : 3 (b) 3 : 1 (c) 1 : 9 (d) 9 : 1

510. In the given figure XY ‖ AC, and XY divides the triangle into two equal parts. Then

(a)

(b)

√ (c)

√

√ (d)

√

√

511. ABCD and DEC are a square and an equilateral triangle respectively. Then ∠DAE is

(a) 450 (b) 300 (c) 150 (d) 22

512. In the given figure AM ⊥ BC and AN is the bisector of ∠A. Then ∠MAN will be- (If ∠B = 650, ∠C = 33



(a) 33

(b) 16

(c) 160 (d) 320

513. What will be the number of diagonals in a regular hexagon? (a) 6 (b) 4 (d) 11 (d) 9 514. What will be the ratio of the sum of the interior angles and that of the exterior angles of an octagon? (a) 1 : 2 (b) 1 : 3 (c) 2 : 3 (d) 3 : 1 515. In the given figure AB ‖ CD then ∠FXE is

(a) 30 (b) 50 (c) 60 (d) 80 516. The remainder after dividing 2x3 – 3x2 + 4x – 1 by (x – 1) will be (a) – 10 (b) 2 (c) – 1 (d) 10

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