12. Circles

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 $AD$`A``D`, and $B$`B` is the point at which the tangent meets the circle.

a

What can we say about the lines $OB$`O``B` and $OC$`O``C`?

$OB=OC$`O``B`=`O``C`

A

$OB>OC$`O``B`>`O``C`

B

$OB`O``B`<`O``C`

C

b

What point on $AD$`A``D` is closest to the center of the circle?

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

d

Hence, what can we say about angle $\angle OBA$∠`O``B``A`?

Straight angle

A

Acute angle

B

Reflex angle

C

Right angle

D

Easy

Approx 2 minutes

In the figure below, $AC$`A``C` is tangent to both circles.

In the diagram, $AC$`A``C` is a tangent to the circle with center $O$`O`. What is the measure of $x$`x`?

If $\overline{BA}$`B``A` is a tangent to the circle, determine the value of $x$`x` showing all steps of working.

Sign up to access Practice Questions

Get full access to our content with a Mathspace account

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.