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Middle Years

9.11 Tangents to a circle

Worksheet
Tangents to a circle
1

In the diagram, C is an arbitrary point on the line AD, and B is the point at which the tangent meets the circle.

a

What can be said about the lengths of OB and OC?

b

What point on AD is closest to the centre of the circle?

c

In general, what can be said about the angle of a line joining some point to some other line by the shortest route?

d

Hence, what can be said about angle \angle OBA?

2

Consider the following figure:

a

Show that \angle OBP is a straight angle.

b

Hence what can we say about points O, B and P?

3

Find the value of the pronumerals in the following diagrams. Give reasons for your answer.

a
b
c
d
e
f
4

The circle in the following diagram has a radius of 8 and AB has a length of 9 units.

Calculate the length of FE.

5

In the diagram, TU and TV are tangents to the circle with centre O. Find the size of \angle SUV.

6

In the following diagram, JL is a tangent to the circle with centre O. JO = 29 and JK = 20. Find the length of the radius OK.

7

In the following diagram, PA and PB are two tangents issued from point P to the circle of center O, such that \angle PBO=37.

a

Find the value of y.

b

Find the value of x.

8

Find the value of the pronumerals in the following diagrams. Give reasons for your answer.

a
b
c
d
e
f
9

In the following diagram, PA and PB are two tangents issued from point P to the circle of center O, such that PB=5.

a

Find the value of x.

b

Find the value of y.

10

In the diagram, PT is a tangent to the circle. Find the size of \angle YZT.

11

In the figure, O is the center of the circle, PA and PB are tangents. Find the size of \angle APB if \angle OAQ=56\degree.

12

In the diagram, PT is a tangent to the circle with centre O. Determine the value of b.

13

In the following diagram, AB, BC and AC are tangents to the circle. BD = 7 \text{ cm}, AF = 6 \text{ cm}, and BC = 11 \text{ cm}. Find p, the perimeter of \triangle ABC.

14

In the diagram, PT is a tangent to the circle. Determine the value of f.

Proofs
15

Prove that \triangle OAC and \triangle OBC are congruent. Then, show that AC = BC.

16

For the following diagram, prove that \\ y + z = x.

17

For the following diagram, prove that \\ AB \parallel FG.

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