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General atg. s Specialist Circle Ge • etry Mensuration Student Booklet Contents: Maths OnLine Lessons & Work Sheets 1. Definitions and Terms 2. Chord Properties 3. Angles at the Centre 4. Angles on the same Arc 5. Cyclic Quadrilaterals 6. Angles in the Semi Circle 7. Tangent Properties 8. Mensuration Work Sheet
Transcript

General atg. s Specialist

Circle Ge • etry Mensuration

Student Booklet

Contents: Maths OnLine Lessons & Work Sheets 1. Definitions and Terms 2. Chord Properties 3. Angles at the Centre 4. Angles on the same Arc 5. Cyclic Quadrilaterals 6. Angles in the Semi Circle 7. Tangent Properties 8. Mensuration Work Sheet

Definitions and Terms - Worksheet

Draw a clear sketch to illustrate the following information in each question, using circles with centre 0 unless otherwise instructed.

Q1 A, B and C are concylclic.

Q2 Shade in the minor segment formed by the chord XY.

Q3 The arc PQ subtends an angle of 100° at the centre.

Q4 The arc IVIN subtends an angle of 40° at a point P on the circumference.

Q5 AB and CD are two chords intersecting at X.

Q6 PQRS is a cyclic quadrilateral.

Q7 ST is a tangent touching the circle at T. SAB is a secant cutting the circle at A and B.

Q8 AB is a common chord to two circles with centres 0 and C respectively.

Q9 PTQ is a common tangent to two circles, centres 0 and C, that touch externally.

Q10 The diameter AB subtends an angle at a point C on the circumference.

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Chord Properties - Worksheet

Give answers correct to 1 decimal place where necessaly.

Q1 Q5 A chord AB is drawn 7cm from the centre of a circle with radius 12cm. Calculate the length of AB.

Q2

Q3

Q4

AB = 8cm. C is the midpoint of AB. If the circle has radius 5cm, find the length of OC.

The chord XY is 24cm long and is 5cm from the centre 0. Calculate the radius of the circle.

OC = 9cm. Find the length of AB.

A chord is drawn 24cm from the centre of concentric circles with radii 26cm and 30cm. Find the length of PR.

Q7

PAB is an isosceles triangle inscribed in a circle of radius 26cm. M is the midpoint of AB. If PA = PB and AB = 48cm, calculate the length PM.

Q6

ANSWERS

A) 3 cm

B) 4 cm

C) 4.8 cm

D) 5 cm

E) 6cm

F) 7.2 cm

G) 7.8cm

H) 8 cm

I) 8.4 cm

J) 9.2 cm

K) 9.7 cm

L) 10.1cm

M) 10.7cm

N) 12 cm

0) 13 cm

P) 15 cm

Q) 16.8 cm

R) 18 cm

S) 19.5cm

T) 21.4cm

U) 22 cm

V) 28 cm

W) 30 cm

X) 32 cm

Y) 36cm

Z) 44 cm

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2

Angles at the Centre - Worksheet

Find the value of x in each diagram.

B

Q5

A0//CB and LA = 24 ° . Find x.

ANSWERS A) 17 °

B) 30°

C) 34°

D) 38°

E) 41 °

F) 45°

G) 46°

H) 50°

I) 52°

J) 58°

K) 70°

L) 72°

M) 82°

N) 108 °

0) 110°

P) 112°

Q) 116 °

R) 124°

S) 134°

T) 138 °

U) 148 °

V) 152°

W) 156 °

X) 164°

Y) 170°

Z) 220°

Q1

Q2

Q3

Q4

Q6

Q7

Q8

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3

Q5 AABC is equilateral.

Q6

Q8

Angles on the Same Arc - Worksheet

Find the value of x in each diagram.

ANSWERS

A) 28 °

B) 30 °

C) 32 °

D) 35 °

E) 36 °

F) 38 °

G) 40 °

H) 42 °

I) 45 °

J) 46°

K) 48 °

L) 50 °

M) 51 °

N) 53 °

0) 54°

P) 58°

Q) 60 °

R) 72 °

S) 80 °

T) 84°

U) 90 °

V) 93 °

W) 97 °

X) 98 °

Y) 102 °

Z) 105°

Q1

Q2

Q3

Q4

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4

Cyclic Quadrilaterals - Worksheet

Find the value of x in each diagram.

ANSWERS

A) 28 °

B) 30 °

C) 31 °

D) 34 °

E) 35 °

F) 38 °

G) 40 °

H) 42 °

I) 51 °

J) 54 °

K) 57 °

L) 60 °

M) 62°

N) 65 °

0) 67°

P) 72 °

Q) 74°

R) 75 °

S) 85 °

T) 86 °

U) 92 °

V) 94 °

W) 105°

X) 115 °

Y) 121 °

Z) 136°

Q1 (X

Q2 Q6

Q3 Q7

Q4 el0

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5

Qf

Q6

Q7

Q8

Angle in a Semi-Circle - Worksheet

Find the value ofx in each diagram, correct to 1 decimal place where necessary.

ANSWERS A) 26°

B) 27°

C) 28°

D) 30 °

E) 32°

F) 38 °

G) 42°

H) 43°

1)44°

J) 45 °

K) 46°

L) 48°

M) 54°

N) 59 °

0) 85°

P) 90 °

Q) 92 °

R) 115°

S) 4.6cm

T) 5.2cm

U) 6cm

V) 7.4cm

W) 8cm

X) 9cm

Y) 9.8cm

Z) 11.1cm

Q1

Q2

Q3

Q4

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6

Tangent Properties Worksheet

Find the value of the pronumeral in each diagram, correct to 1 decimal place where necessary.

Qi 20cm Q5

Q6

68' 0 •

ANSWERS

A) 4 cm N) 16.8cm

B) 4.7 cm 0) 17.7cm

C) 5 cm P) 17.9 cm

D) 5.1 cm Q) 18.2cm

E) 7 cm R) 21.5 cm

F) 8 cm S) 23.3 cm

G) 8.9 cm T) 24.2cm

H) 9.3 cm U) 74 cm

I) 10 cm V) 80 cm

J) 10.6 cm W) 82 cm

K) 11 cm X) 44°

L) 11.5 cm Y) 64°

M) 15.5 cm Z) 68°

Q3 PT is a tangent to the circle touching it at T. PT = 8cm and PO = 12cm. Find the length of diameter AB.

Q7

10cm

0

7cm

OC = 18cm. Find x.

Q4 12cm

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I. Calculate, in degrees(2 dit.t....4), the magnitude of the angle subtended at the centre of a circle of radius length 8 cm by an arc whose length is 15 cm.

2. Find the length of an arc of a ci'rcle of radius 15 cm if the arc subtends an angle of 70° at the centre.

3. An arc of a circle subtends an angle of 52 ° at the circumference. lithe radius length of the circle is 12 cm, calculate the arc length.

4. The area °fa sector OA B of a circle, centre 0, radius length 20 cm is 240 cmz. Calculate (a) the magnitude or the angle 408, (b) the length of arc AB, (c) the length of the chord AB.

5. A chord PQ, 24 cm long, is 5 cm from the centre of the circle. Calculate the length of the arc PQ.

6. A point P is 8 cm distant from the centre of a circle of radius length 5 cm. Find the length of the major arc between the points of contact of the tangents drawn

from P to the circle.

7. A chord AB of a circle with centre 0 has length 16 cm. lithe radius of the circle

is 10 cm, calculate (a) the magnitude of angle A08, (b) the length of the minor arc AB. (c) the area of the minor segment formed by the chord AB.

8. The minute hand of a clock is 20 cm long. Calculate (a) the arc length along which the tip of the hand travels in 16 min. (b) the shortest distance between the initial and final positions of the tip of the

hand.

9. An arc AB subtends an angle of 06.271vat the centre of a circle of radius 15 Ctn.

Calculate the difference in the lengths of the chord AS and the minor arc AB.

10. From a circular piece of metal 6 cm in diameter, a sector of angle 30' is removed. Find the area remaining. Express your answer in terms of n.

II. A chord subtends an angle of 126-).06 at the centre of a circle of radius 16 cm. Find the difference in length between the chord and the arc.

12. Two circles of radii 6 cm and 8 cm have their centres 10 cm apart. Calculate the

length of the arc of the smaller circle cut off by the larger circle.

13. The minute hand of a clock is 3 cm in length. What area is swept out by the hand in an interval of 40 min? Express your answer in terms of n.

14. A circular metal plate is cut into two segments along a chord equal in length to the radius. What is the ratio of the areas of the two segments?

15. A sheep, grazing in a paddock, is tethered to a stake by a rope 20 m long. If the stake is 10 m from a fence, find the area over which the sheep can graze.

16. Two tangents, PA and PB, are drawn to a circle with centre 0 and radius 5 cm from an external point, P. If PO = 13 cm, calculate the area hounded by the tangents and the minor arc AB.

II. A trcel5t-lesSLn ittat

ox-Dv.Aci 5 CtAtTe.S 0.rt

po_sces ) t.,31x110,...t crosst %-isal?) , 6314.--S whi:Dsra cluars art tn. 0,Ack 3. vk ana

3 .1 5 t-'■ 01)0,rt , cta. t..4. at, ta, izkAssti, di;

ttsiz bat. 11 . 111, 0 Cd-a-A-S o, cia ct.U.A.s ,Qsu\sti., k0 (4A ) Leure_ ta,4)5

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C._

Mensuration Work sheet 8

(6) 2-4- c-/-1 Z2 - 6-

C) •55 (c-)

g . (0,) 11 '5 cm (6) 21

M. (21V : IT 4-

tb •

. • 6 crA

i _i.ktt- ' PA

. 12 ."7 0 crAl

Mensuration Work sheet 7

SAC Work Sheet 1-7 Calculator ActiveSAC Work Sheet 1 - 8 Calculator Active

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