# Tangents to Circles

## Interactive practice questions

In this question we aim to prove that the tangent is perpendicular to the radius drawn from its point of contact.

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

a

What can we say about the lines $OB$OB and $OC$OC?

$OB=OC$OB=OC

A

$OB>OC$OB>OC

B

$OBOB<OC C$OB=OC$OB=OC A$OB>OC$OB>OC B$OBOB<OC

C
b

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

Point $A$A

A

Point $B$B

B

Point $C$C

C

Point $D$D

D

Point $A$A

A

Point $B$B

B

Point $C$C

C

Point $D$D

D
c

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

The angle is obtuse.

A

The angle is reflex.

B

The angle is a right angle.

C

The angle is acute.

D

The angle is obtuse.

A

The angle is reflex.

B

The angle is a right angle.

C

The angle is acute.

D
d

Hence, what can we say about angle $\angle OBA$OBA?

Straight angle

A

Acute angle

B

Reflex angle

C

Right angle

D

Straight angle

A

Acute angle

B

Reflex angle

C

Right angle

D
Easy
Approx 2 minutes
Prove that $\triangle OAC$OAC and $\triangle OBC$OBC are congruent. Then, show that $AC=BC$AC=BC.
Prove that $y+z=x$y+z=x.