c hord r a c sector m a j o r m i n o r m a j or m i n o r g e m s e n t di ameter r a d i u s s emi-circle tangent a c r A line segment drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. i.e. AP PB OP AB A perpendicular from the centre of a circle to a chord bisects the chord. i.e. OP AB AP PB A B O P A B O P O P A B T x 4,5 12 O Q R S T M N x 6 8 10 O G H D E x 5 5 2 O A B C D K x 9 8 12 2 3 GEOMETRY CIRCLES Terminology Chord Theorems ¤ Reference: chord theorem either of these can be proven by joining OA and OB and using congruency. Chord Theorem corollary ¤ The centre of a circle lies on the perpendicular bisector of the chords. ** e.g.1 Determine the value of x, with reasons (correct to 1 dec. pl. if necessary): 1. 2. 3. 4.
Transcript
chord
ra csec
tor
major
minor
maj or
min o r ge ms
en t
diam
eter
radius
semi-circ
le
tangent
a cr
A line segment drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
i.e. AP PB OP AB
A perpendicular from the centre of a circle to a chord bisects the chord.
i.e. OP AB AP PB
A
B
O
P
A
B
O
P
O
P
A
BT
x
4,5
12
O
QR
ST M
N
x6
8
10
O
G
H
D
E
x
5
52
OA
BC
D
K x
9
812
23
GEOMETRY
CIRCLES
Terminology
Chord Theorems
¤
Reference: chord theorem either of these can be proven by joining OA and OB and using congruency.
Chord Theorem corollary
¤ The centre of a circle lies on the perpendicular bisector of the chords.
** e.g.1 Determine the value of x, with reasons (correct to 1 dec. pl. if necessary):1. 2.
3. 4.
O
MP
Q
O
QT
P
S
O
MA
B
O
MA
C
B
O
A
BC
D
K x
9
812
23
G
H
GEOMETRY
»Solutions
1. TB AT 6 (chord thm)OP OB (radii)
2 2 2OB 4,5 6 (Pythag. thm)
56,25OP 56,25 7,5
2. OM TSNQ 4; MS 5 (chord thm)OS OQ 6 (radii)
2 2
2 2
OM 6 5
3,317
ON 6 4
4,472
(Pythag. thm)
MN 3,317 4,472 7,8
3. OD GH (chord thm)OE x (radii)OD 2x
2 2 2OD GD OG (Pythag. thm)2 2
2 2
( 2) 25
4 4 25
7,3
x x
x x x
x
§ Exercise 1
1. O is the centre of the circle; OM PQ; OP 50 mm and OM 30 mm.
1.1 Calculate PM.1.2 Give the length of PQ.
2. O is the centre of the circle; SP 100 mm; OTQP and OT 40 mm.Calculate the length of QP.
3. O is the centre of the circle; M is themidpoint of AC; OM 20 mm andAC 40 mm.Calculate the length of AB.
4. AB 80 mm; MO 40 mm and AM MB. Calculate the diameter of the circle.
O is centre.
4.
GB 6
HD 7
(chord thm)
GK 9 6 3
OG HK 12 7 5
2 2 2GK GOx (Pythag. thm)2 23 5 34
34 5,8x
O
A
BM
C DN
C
A BM
O
O
P Q
R S
O
P Q
R S
x
20°
xy
30°40° x
18° 26°
Ox48°
O
C D
A B
O
C D
A B
CA B D
O
A
B
M N
GEOMETRY
5. O is the centre; AB 60 mm; OM 40 mmand ON 30 mm. Calculate the radiusof the circle and the length of CD.
6. If AB 6 units and MC 1unit, and
OMAB, find the length of OM.O is centre.
7. PQ 60 mm; RS 80 mm and the radius of the circle is 50mm. O is the centre.Find the distance between the parallelchords in each case.7.1 7.2
8. CD 10 cm; AB 24 cm and the radius of the circle is 13cm. O is the centre.Find the distance between the parallelchords in each case.
9. Two concentric (same centre) circleswith centre O have radii 17 cm and 10 cm.
If AD 30 cm, calculate the length of BC.
10. Circle M passes through the centre ofcircle N and circle N passes through thecentre of circle M. The circles inter-sect at A and B. Determine the lengthof the common chord AB in terms ofthe radius, r, of the circles.
§ Exercise 2 Find the size of x and y :
1. 2. 3.
4.
O centre
O
124°
B
C
A
x
O
B
AC
x 10°
110°O
x
C
BD
O
70°x
yA
B
C
O
244°
B
C
A
x
D
y
O
BA
C
40° 1
22
x
1
1 2
50° 70°A
C
B
Ox2 OD A
CB
3 30°
50°
xyz
O
A B
P
O
A B
PO
A B
P
O
A B
P
O
AB
P
1
1
2
2
GEOMETRY
Angle at centre
¤ The angle subtended by an arc of a circle at the centre is double the angle subtended bythe arc at any point on the remaining part of the circumference.
Reference: at centre
i.e. O 2P
, O centre
Why?
1 1 2 2
1 2 1 2
ˆ ˆˆ ˆO 2 P and O 2 Pˆ ˆO 2 P
§ Exercise 3 O is the centre of the circle. Determine the sizes of the required angles,
giving reasons.
1. 2. 3. 4.
5. 6. 7. 8
O
B A
C
x
120°
O
B A
C
40° 1
22
x
1
1 2
50° 70°A
C
B
Ox2 OD A
CB
3 30°
50°
xyz
O
A
B
C D
x
70°1
O
A
B
C
x
O
70°
BC
A
x OB
C
A
x
35°
Ox C
BD
Ox
A B
C
O
B A
C
x
115°
O
B A
C
25° 1
2x
O
A
B
P
GEOMETRY
§ Exercise 4 O is the centre of the circle. Determine the value of x, y and z.1. 2. 3. 4.
5. 6. 7. 8
9. 10. 11. 12.
Angle in semi-circle
As illustrated in no.10 in the last exercises, the angle subtended by a diameter is 90°.
¤ The angle in a semi-circle is a right angle.
Reference: in semi-circle
Why?
ˆ ˆˆO 2 P and O 180
P 90
70°
x
y zO
O
A B
P Q
GEOMETRY
Angles in the same segment
§ Exercise 5 O is the centre of the circle. Determine the sizes of the required angles.
¤ The angles in the same segment of a circle are equal.
Reference: 's in same segment
Why?
ˆ ˆ ˆˆO 2 P and O 2 QˆP Q
Angles in same segment corollaryA corollary, is a statement that follows readily from a previous statement/theorem.
¤ Angles subtended by equal chords are equal.
Reference: 's subt. by = chords
Hopefully obvious fromthe previous theorem?
B
A
C
D
180°
B A
CD
180°
O
AB
D
C
1
2
AB
D
C
2 1P
GEOMETRY
Cyclic quadrilateralsi.e. quadrilaterals with vertices on a circle.
¤ The opposite angles of a cyclic quadrilateral are supplementary.
Reference: opp. 's cyclic quad.
i.e. A C 180 ; B D 180
Why?
1 2
1 2
ˆ ˆ ˆ ˆO 2 A and O 2 Cˆ ˆ ˆ ˆO O 2 A C 360
ˆ ˆA C 180
¤ If one side of a cyclic quadrilateral is produced the exterior angle formed is equal to the interior opposite angle.
Reference: ext. cyclic quad.
Why?2 1 2
1
ˆ ˆ ˆ ˆD B 180 and B B 180ˆ ˆD B
A B
180°
180°
GEOMETRY
SummaryReference: at centre
i.e. O 2P
, O centre
Reference: in semi-circle
Reference: 's in same segment
Reference: 's subt. by = chords
Reference: opp. 's cyclic quad.
i.e. A C 180 ; B D 180
Reference: ext. cyclic quad.
A
BC
D
x
yz
100°
P
S
Q
Rx
y
z70°
80°
55° T
S
R
Q
xO
Py
z
120°
P
L N
M
J
x
y
z
w
110°
Mx
35°
101°
y
z
A
B
DC
O
61°x
z35°
P
Q R
S
T
B
CP
Q
A
50°
x
z
R
y
w
A
B
F E
D
C
34°x
y
O z
A
BE
O
D
C
30°
x
A
O
BC
E
x
75°
A
B
O
C
D
E
80°
x
O
A
B
C
D
70°
x
A
B
C
D
E
O
120°
x
OA
B
C
D
E
120°x
35°
D
GEOMETRY
§ Exercise 6 Find the values of w, x, y and z, giving reasons.
1. 2. 3.
4. 5. 6.
7. 8.
§ Exercise 7 Calculate the value of x.
1. 2. 3.
4. 5. 6.
D
A
C
B
O A
C
D 30°
x
O
A
B
C
E
70°
x O
AB
C
D
E55°
x
O
AB
C D
70° xO
A
B
C
D
E
x
70°
O
A
B
C
D
120°
x
O
A
B
C
DE
x
70°
O A
BC
D
E
65°x25°
O
ABC
D
E
30°x
75°
O
A
B
C
D
E
110°
x
FO
A
B
C
D110°
x
O
A
B
CD
70°
x
25°
GEOMETRY
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
Concyclic points
Points are concyclic if they lie on a circle.
In the figure A, B, C and D are concyclic.
It means the same to say that ABCD is acyclic quadrilateral.
i.e.
i.e.
x
180°-x
x
180°-x
i.e.
P
QR
S
T
40°
75°
35°
21
PQ
R S
M 110°
120°
50°
A
BC
O
D
E
1
1
1
1
2
2 2
2
A
BC
D
P
50°
1 2
GEOMETRY
¤ If the line segment joining two points subtends equal angles at two other points on the same side of it, then the four points are concylic.
Reference: conv.'s in same segm.
¤ If one pair of opposite angles of a quad-rilateral is supplementary then the quad-rilateral is cyclic.
Reference: conv. opp.'s cyclic quad.
¤ If one side of a quadrilateral is produced and the exterior angle formed is equal to the interior opposite angle, then the quad-rilateral is cyclic.
Reference: conv. ext. cyclic quad.
§ Exercise 8
1. Prove that PQRS is a cyclic quadrilateral in the following examples, giving reasons:1.1 1.2
2. BDAC; AEBC; AE intersects BD at O. Name two cyclic quadrilaterals in this figure,giving reasons for your answers.
3. APC and BPD are straight lines, BPAP,AB//CD, B 50 .
Prove, with reasons,
that A, B, C and D are concyclic.
C
A
B
DE
F
K
12
21
1
65°
Given O centreR.T.P OPABConst. OA, OBProof In Δ’s OAP, OBP
1. OAOB (radii)2. APPB (given)3. OP commonΔOAPΔOBP (SSS) 1 2P P
(Δ’s proven)
But 1 2P P 180
(adj. 's st. line)
1 2P P 90
A
B
O
P1 2
GEOMETRY
4. Prove, with reasons, that B, C, F and K are concyclic.
Proofs of some theorems You will need to know these:
Chord Theorems
A line segment drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
A perpendicular from the centre of a circle to a chord bisects the chord.
You should be able to complete this proof yourself.
Given
R.T.P.
Const.
Proof
21
P
A
O
B
Given O centre
R.T.P. O 2P
Const. Join PO, produce to N
Proof 1 1O A P
(ext. Δ)
OAOP (radii)
1A P
(isos Δ)
1 1O 2P
Sim’ly 2 2O 2P
1 2 1 2
1 2 1 2
in : O O 2P 2P
in : O O 2P 2P
ˆ ˆO 2P
O
AB
P
N
1
1
2
2
O
AB
PN
1
22 1
R.T.P. A C 180 ; B D 180
Const. O centreJoin BO, DO
Proof 1O 2A
( at centre)
2O 2C
( at centre)
1 2O O 360
( 's at a pt)
2A 2C 360
A C 180
Sim’ly, by joining AO, CO, it can be proven thatB D 180
B
A
C
D
O 12
Given ˆ ˆA C 180 R.T.P. ABCD is a cyclic quadAssume that ABCD is not cyclic. Const. Let the circle thro’ B, C and D
meet BA, or BA produced, at P.Join PD.
Proof ˆ ˆA C 180 (given)
1ˆP C 180 (opp. 's cyclic quad.)
1ˆ ˆA P
But this is impossible, since,
1 1ˆ ˆ ˆin , A P D (ext. )
1 1ˆˆ ˆin , P A D (ext. )
the original assumption is falseABCD is a cyclic quad.
B
A
C
D
P1
1B
A
C
D
P
1
11
GEOMETRY
The angle subtended by an arc of a circle at the centre is double the angle subtended bythe arc at any point on the remaining part of the circumference.
The opposite angles of a cyclic quadrilateral are supplementary.
If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.
This is a method known as“reduction ad absurdum”(reduction to the absurd)i.e. proof by contradiction
A
B
C
D
P
Thought strategy:What if AB//CD? ... A D
Prove A D.
To do this:
What is A to?
... B
first prove B D.
Prove that AB//CD.
R.T.P AB//CD
Proof A B (isos )
B D ( 's in same segm.)
A D
AB//CD (alt. 's )
Thought strategy:What if BCFK is cyclic? ... 2 1B F
2 1Prove B F .
To do this:What is 2B to?
... 2D
1 2first prove F D .
Prove that BCFK is a cyclic quadrilateral.
R.T.P BCFK cyclicProof 2 1D F (ext. cyclic quad.)
2 2B D (ext. cyclic quad.)
2 1B F
BCFK cyclic (conv. ext. cyclic quad.)
C
A
B
DE
F
K
12
21
1
2
Thought strategy:What is 1B to?
... ???What does 1B
combine with?
... sum .
Prove that 1B C
.
R.T.P 1B C
Proof 1A B 90 (ext. )
A C 90 ( sum )
1B C ( sum )
D
A B
C
1
1
GEOMETRY
Geometric riders (proofs)When required to prove a geometric fact it is always good to have a thought strategy .
The most important overriding factor to any strategies ... know your theorems.One useful strategy:
If required to prove something about lines or shapes (e.g. prove ACB is a straight line ... prove AB//CD ... prove ABCD is a parallelogram ... prove ABCD is a cyclic quadrilateral... prove that ABCD), ask “What if this is so?” This will usually lead on to another fact which may be easier to prove first (as long as the converse is true!).
If required to prove two angles equal (and sometimes two sides), take one and ask “What is this angle (or side) equal to?” and then see if the other angle can be proven equal to the same angle.
If required to prove two angles equal and the previous strategy does not work, try angle combinations (e.g. sum )
**e.g.2
**e.g.3
**e.g.4
A
B
C
D
E
2
2
1
1
DA
B
C 1
1
11
2
2
2
2
A
C
PB
Q
1 2 3
P
QMA
1 2 3
OB
A
D
P
G
1 2
QF
PB
Z2
1
A
B
C
PQ
11
1
12
2
22
3
3
A DB C
P Q SR1
1
1
12
2 2
2
D
G
F
B
AC
E
1
1
1
1
12
2
2
2
2 3
3 4
D B
C
AE
O
11
1
1
22
2
3
GEOMETRY
§ Exercise 9
1. 2. 3.
A B
ADAB ABAC
Prove: 1.1 CD//AB Prove that AC bisects C.
AP//BQ1.2 ECED Prove that AQ//CP
4. 5. 6.
PMAB; PQ is a diameter. AP and PD are diameters. FP is a diameter, QF//BP.Prove that 1 3P P .
Prove that A, G and D are Prove that QB is a diameter.
collinear.
7. 8.
ABCD and PQRS are straight lines.
1 1B C
Prove: 8.1 AP//CR
Prove that 1Q ABC
8.2 APSD is a cyclic quadrilateral.
§ Exercise 10
1. Prove that GCBD is a cyclic quadrilateral. 2. Prove that AODC is a cyclic quadrilateral.
While there is a rigorous (formal) proof for this theorem, the easiest way to see it is in terms of the shortest distance from a point to a line, the perpendicular. The radius is obviously the shortest distance from the centre to the tangent ... 90
Why?
i.e.
D B
AE
O
1
1
2
2
C
F D
B
A
E
O
C
12
B
A
C
D
x
y1
2
1
B
C
D
pA E
q
r
GEOMETRY
3. Prove that A, B, C and F are concyclic. 4. Calculate the value of x, with reasons.
5. Find the relationship between x and y, 6. Calculate the value of .p q r with reasons.
Tangents to circles A tangent touches the circle.
¤ The angle between a tangent to a circle and the radius drawn from the point of contact is a right angle.
Reference: rad. tan.
¤ If a tangent is drawn to a circle and a chord is drawn from the point of contact, then the angles between the tangent and chord are equal to the angles in the alternate segment.
Reference: tan.chord (or alt. segm. thm or in the alt. segm.)
= = by tan. chord, meaning that the tangents are equal, by isos Δ.
Why?
Cx
A
B
y
z
67°
DB
C
50°
A
70°
x
Y
xO
y
z
150°
P
S
TK O is centre.
M
40°
x
N
G
y
L
12
S
x
F
72°
TG
P
64°
y
zE
1A
P
T
B
F Gx y
1
2
30°
With the diameter drawn in from the point of contact, we have two right angles, one by rad. tan. and the other by in semicircle.
Now, = by ’s in same segm., leaving the remainder of the right angles, , equal.
Why?
GEOMETRY
¤ Tangents from a common point to a circle are equal.
Reference: tans from common pt
§ Exercise 11
Find x, y and z, giving reasons:
1. 2. 3.
4. 5. 6.
T110°
x
z
25°
A
J
PQ
RB
y
A 80°
x
RB
US
D T C80°70°
y
zw
1
11
1
2
22
2
3
3
3
4
40°
x
y
A
B
C
D
BA is a tangent.
1
42°
x
y
S
P O
T
D
O is centre; PS, PT are tangents.
1
1
x
120°
y
z
1
12
23
O
A
B
C
DP Q
O is centre.
A
B
C
Ox
y62°
D
1
1 2
A
B
C
O
55°
x
12
A
B
C
O x
76°1
A
B C
Ox
35°
1
12
A
B
CO x 40°
A
BC
O
x
66°
D
1 2
A
B C
O
x
40°
1
12
B
C
O
x 50°A
A
BC
O x
130°
C
D
12
P
GEOMETRY
7. 8. 9.
10. 11.
§ Exercise 12 Calculate the values of x and y, with reasons.
1. 2. 3.
4. 5. 6.
7. 8. 9.
A
B
C
D
A
B
C
D
ˆ ˆB D AB is a tangent to circle BCD.
A
BC O
xy
54°
P
Q
1
A
B
O
x
y
50°C
D
1
12
A
B
C
O
x
y
80°
D
E1
2
2
1
D
E
G
F
B
C
A
1
1
1
1
1
2
2
2
2
23
3
D
B
C
W
S
A
T1
1
1
1
42
2
2
22
3
3
1
5
GEOMETRY
10. 11. 12.
¤ Converse to alternate segment theorem
Reference: conv.tan.chord
§ Exercise 13
1. Two circles intersect at A and B. Line DE is a tangent to circle DABC and line BD is a tangent to circle BAGF.
Prove that:
1.1 CF//DE1.2 ED is a tangent to circle EAG.
2. TD is a tangent to the circle ABCD, AD//BC, DC and AB produced meet at W, and TBS is a straight line.
ˆ ˆWBT CBD.
Prove that:
2.1 BWTD is a cyclic quadrilateral.2.2 TBS is a tangent to circle ABCD.
C
P
B
D
A Q
1
2
x 3
4
y
A
T
P R
B
Q 1
1
1
2
2
3
2
34
A
BC
Q
D
P1
2
2
2
3
11
C
D
EA
T
B1
1
1
1
22
2
2 34
AB
CP
T1
23
GEOMETRY
3. PAQ is a tangent to the circle,
with ˆPAD y and ˆAPD .x
3.1 Express all the numbered angles in the figure in terms of x and/or y.
3.2 What is the relationship between x and y if AB is the diameter?
4. Two circles intersect at P and Q. Chord PA is a tangent to the smaller circle, chord PB is a tangent to the larger circle, and AB intersects the circles at points D and C.
Prove that PD PC.
5. PA and PB are tangents to the circle. PQRS, a straight line, is parallel to chord BT; AB andAR are drawn.
5.1 If T ,x name four other angles each equal to x. Give reasons for your answers.
5.2 Prove that PARB is a cyclic quadrilateral.
5.3 Deduce that RB RT.
§ Exercise 14
1. PCT is a tangent to the circle at C.AB//PT.Prove that AC BC.
2. EAT is a tangent at A and AD AB.Prove:2.1 DB//EAT
2.2 3ˆ ˆA E
2.3 1 2ˆ ˆC C
1
1
2
2 3
P
Q
R S
T
U
1
2
2
2
2
3 3
341
11
D
A
EC F
B
1
1
1
1
12
2
2
2
22
3
3 3
3
4
4
4
41
B
P
A
CE
D
D
C
A F P
B
1
1
11
1
1
22
22
2
2 3
3
43
4
1
11
1
1
2
22
22 3
3
4 N
RQ
M
P
S
T
A
B
C
D
E
1
1
1
1
2
2
2
2
3
4
T
GEOMETRY
3. QT and RS are parallel chords;TS is produced to U.Prove:
3.1 1ˆ ˆS P
3.2 2ˆ ˆQ T
4. ECF is a tangent to the circle at C,and AB//EF. Prove:4.1 AC BC
4.2 3ˆE C
4.3 1 4ˆ ˆ ˆA C F
5. BP and AP are tangents, and BP AD.Prove:5.1 BDP ADP 5.2 1 2
ˆ ˆC C5.3 BE EA5.4 BA CP
6. PA and PC are tangents to the circle at A and C. AD//PC, and PD cuts the circle at B. CB is produced to meet AP at F. AB, AC and DC are drawn.Prove:6.1 AC bisects ˆPAD
6.2 1 3ˆ ˆB B
6.3 ˆ ˆAPC ABD6.4 4 2
ˆ ˆA P
7. MN is a diameter, and PQ MN.Prove:7.1 TSRN is a cyclic quadrilateral.
7.2 1 1ˆ ˆS N
7.3 MP is a tangent to the circlethrough PTN.
8. ABCD is a cyclic quadrilateral withthe tangent at A parallel to BD.Prove:8.1 AB AD
8.2 AC bisects ˆBCD8.3 AB is a tangent to circle BCE.
O
A
B
D
E
1
1
1
2
2
2
3
C
1
1
1 1
1
1 2
2
22
2 2334 4
D
B
A
C
L
T
M N
K
GEOMETRY
9. O is the centre of the circle. AC and EC intersect the circle at Band D respectively, and AC EC. Prove that AODC is a cyclicquadrilateral.
10. AB and CD are two chords thatintersect at T. AK CD and DL AB.10.1 Prove:
10.1.1 AKLD is a cyclic quadrilateral.
10.1.2 KL//CB10.2 If AK and DL produced cut CB
at M and N respectively, prove thatAMND is a cyclic quadrilateral.
R.T.P. 1 1 2A C , A D
Const. Diameter ATJoin TC
Proof In fig.1:
1 2A A 90 (rad.tan.)
1 2C C 90 ( in semi-circle)
But 2 2A C ( 's in same segm.)
1 1A C
In fig.2:
C D 180 (opp. 's cyclic quad.)
1 2A A 180 (adj. 's st line)
2A D
C
A
B
PO
T
12
12
fig.1
C
A
B
PD
12
fig.2
Given ˆ ˆPAB CR.T.P. PA is a tangent to circle ABC.Assume that PA is not a tangentConst. This means that tangent QA can be
drawnProof QAB C (rad. tan.)
But PAB C (given)
But this is impossible as they share aCommon arm AB, on the same side ofAB. the original supposition is falsePA is tangent to circle ABC
C
A
B
PQ
GEOMETRY
Proofs of theorems You will need to know these:
¤ If a tangent is drawn to a circle and a chord is drawn from the point of contact, then the angles between the tangent and chord are equal to the angles in the alternate segment.
Reference: tan.chord (or alt. segm. thm or in the alt. segm.)
¤ A line which forms an angle with a chord of a circle, at its point of contact with the circle, equal to the angle subtended by the chord in the alternate segment is a tangent to the circle.
Reference: conv. tan.chord (or conv. alt. segm. thm or conv. in the alt. segm.)
ANSWERS
Exercise 11. 40 mm; 80mm
2. 60 mm 3. 40 2 mm 4. 80 2 mm5. 50 mm; 80mm 6. 4 units 7. 7.1 10 mm 7.2 70 mm
