Gr 10 - Circle Geometry August 2015

Name: ........... ...................... ........................................……….........

Section A

1. Find the values of the angles indicated with a letter in the following diagrams. O always represents

the centre of the circle. You are not required to show any working; write your answers in the grid at the bottom of the page.

a

b

c

d

e

f

g

h

i

j

k

l

m

n

p

q

r

s

t

u

v

w

x

2. Find the values of the angles indicated with a letter in the following diagrams. O always represents

the centre of the circle. You are not required to show any working; write your answers in the grid at the bottom of the page.

a

b

c

d

e

f

g

h

i

j

k

l

m

n

p

q

r

s

t

u

v

w

x

3. Find the values of the angle indicated x in the following diagrams. O always represents the centre of the circle. You are not required to show any working; write your answers in the grid at the bottom of the page.

1

2

3

4

5 6

7

8

9

10

11

12

13 14

15

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15

4. Find the values of the angle indicated x in the following diagrams. O always represents the centre of the circle. You are not required to show any working; write your answers in the grid at the bottom of the page.

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8

5. Find the values of the angles indicated with a letter in the following diagrams. O always represents

the centre of the circle, and radii do indeed meet the tangent at the point of contact. You are not required to show any working; write your answers in the grid at the bottom of the page.

a

b

c

d

e

f

g

h

i

j

k

l

m

n

p

q

r

s

t

u

v

w

x

y

z

a

b

c

d

e

f

i

k

m

y

z

Section B (Finding Angles) In these diagrams, lines that appear straight are indeed straight. O refers to the centre of the circle when

it appears. You must use the information given and fill in the sizes of ALL the other angles.

1.

2. Δ ABC is equilateral

3. TB = TA

4. TOP is a diameter and TA is a tangent.

Section C (Deducing Angles) In the following questions you should show your argument properly written out. on these pages 1. Find x

2. Find x

3. Find x

4. Find x if ˆBAC = 128°

5. Find x and y

6. Find x

7. Find x

8. Find x

9. Find x

10. Find x

11. Find x

12. Find x

13. Find FOUR other angles equal to θ

14. Find FOUR other angles equal to θ

15. TA is a tangent to the circle. AE and

CD are equal chords, and EB TA.

If ˆTAE = x, find, with reasons, five

other angles equal to x

16. ABCD is a cyclic quadrilateral with AB = BC.

AD is produced to E so that CE BD.

FA and FB are tangents at A and B, and ˆDCE = x.

16.1 Name, with reasons, four other angles equal to x

16.2 Prove that ˆAFB = ˆABC

16.3 Name, with reasons, another angle in the figure which is equal to ˆAFB

Section D (Proofs) In the following questions you should show your argument properly written out. on these pages

1. If AC is a diameter and OP bisects AB,

prove that OP || BC

2. If AB = AC, prove that BC || TA

3. PQB is a tangent, and SQ bisects ˆBQR

Prove that QS = RS

4. AB is a tangent at B to the circle with centre O,

and AC cuts the circle at E; D is the midpoint

of CE.

Prove that OBAD is a cyclic quadrilateral

5. ABC is a tangent to the circle at B; BE || CD

Prove that 2ˆD =C

6. Prove PQ || AB

7. Prove BPQC is a cyclic quadrilateral

8. ABCD is acyclic quadrilateral, and AD bisects ˆEAC

Prove that DC = DB

9. AQ is a tangent, and QP || AR

Prove that RA is a tangent to the circle ABQ

10. PR is a diameter and OM bisects ˆSPR Prove that

10.1 PS || OM 10.2 OM ┴ SR

10.3 OM bisects SR

11. TN is tangent at T, O is the centre, LN ┴ NP.

Prove that

11.1 MNPT is a cyclic quadrilateral

11.2 NP = NT

12. PC bisects ˆDCB .

Prove that PA bisects ˆXAB

13. If ALB is a tangent, and parallel to MP, prove 13.1 LM = LP

13.2 LN bisects ˆMNP 13.3 LM is a tangent to the circle MNQ

14. TA is a tangent, TM = PM and TA ┴ PA. Prove 14.1 MTAR is a cyclic quadrilateral 14.2 PR = RT

14.3 TR bisects ˆPTA

14.4 2 1

1 ˆT = O2

Section E (More Angles)

1. Given that P = 51 and O is the centre of the circle,

Find the sizes of the following angles, giving reasons:

1.1 ˆTAO

1.2 ˆAOB

1.3 ˆAQB

1.4 ˆBAT

1.5 ˆOBA

2. Δ PRS is equilateral. Calculate

the values of x and y

3. M is the centre of the circle.

ˆBDE = x.

Express in terms of x (giving reasons)

ˆBME =

ˆBEM =

ˆCAD =

4. If the chords QR and AR are equal in length,

prove that QR is a tangent to the circle through

Q, P and T by proving that ˆ ˆTQR = P

5. In the figure, M is the centre of the circle.

EA = ED ˆAMD = 100

Calculate the measure of ˆABE

6. ABCD is a cyclic quadrilateral and FAS is

a tangent meeting CB produced in F.

AD is produced to E.

ˆCDE = y and ˆACB = x

6.1 Give, with a reason, an angle equal to x

6.2 Give, with reasons, two angles each equal to y

6.3 Express F in terms of x and y

6.4 Express ˆBAC in terms of x and y

6.5 What value of y will ensure that AC is a tangent to the circle AFB

(i.e. through the points A, F and B)?

D

A

B

X

C

T

118

43

1

2

34

5

6

7

P

S

R

T

Q

1

1

65

1

2

2

2

3

1

17

2

2

7. DCX is a tangent to the circle.

TC = TB.

Find the following angles:

1T

2B

3C

4C

5A

6B

7D

8. TS = TR. PQ and PS are tangents.

PSQ = 65° and TSQ = 17°.

Find the size of the following angles:

R

1S

3T

1T

2T

1Q

2S

2P

2Q

1P

PICTORIAL SUMMARY OF CIRCLE THEOREMS

2

180°-

The angle subtended by a

chord at the centre is twice the

angle it subtends at a point on

the circumference

Angles subtended by the same

chord in the same segment are

equal

The angle subtended at the

circumference by a diameter is

a right angle

An exterior angle of a cyclic

quadrilateral is equal to the

opposite interior angle

A radius perpendicular to a

chord bisects the chord; the

perpendicular bisector of a

chord passes through the

centre

The two tangents to a circle

from a point outside the circle

are equal in length

radius perp to chord OR

radius bisects chord

angle at centre

angles in same segment

tangents from same

point

subt by diameter

ext angle of cyclic quad

ARROWHEAD

DONKEY’S EARS

The angle between a tangent

and a chord through the point

of contact is equal to the angle

subtended by the chord in the

alternate segment

tan-chord theorem

WINDSURFER

Equal chords in a circle

subtend equal angles

subt by equal chords

The opposite angles of a

cyclic quadrilateral are

supplementary

opp angles of cyclic

quad