Hong Kong

Stage 4 - Stage 5

Lesson

A tangent is a line that intersects the circumference of a circle in exactly one point, which we call the point of tangency.

A tangent is perpendicular to the radius from the point of tangency. Conversely, the perpendicular to a radius through the same endpoint is a tangent line.

There can be more than one tangent on a circle. In fact there is basically an infinite number! The diagram below shows two tangents- $PM$`P``M` and $PQ$`P``Q`.

If two tangents are drawn from a common point, the tangents are equal.

*Proof:*

Let's start by drawing in radii from the points of tangency:

In $\triangle OMP$△`O``M``P` and $\triangle OQP$△`O``Q``P`:

$OM=OQ$`O``M`=`O``Q`(radii in a circle are equal)

$OP$`O``P` is common

$\angle OMP=\angle OQP$∠`O``M``P`=∠`O``Q``P`$=$=$90^\circ$90° (tangents meet radii at right angles)

$\therefore$∴ $\triangle OMP$△`O``M``P`$\cong$≅$\triangle OQP$△`O``Q``P` (RHS)

$\therefore$∴ $MP=QP$`M``P`=`Q``P` (corresponding sides in congruent triangles are equal)

Remember, we aren't limited to the rules of circle geometry. We can use all our geometrical rules, including Pythagoras' theorem, congruency and similarity.

Let's look through some worked examples now to see this in action.

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.

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`CWhat point on $AD$

`A``D`is closest to the centre of the circle?Point $A$

`A`APoint $B$

`B`BPoint $C$

`C`CPoint $D$

`D`DIn 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.

AThe angle is reflex.

BThe angle is a right angle.

CThe angle is acute.

DHence, what can we say about angle $\angle OBA$∠

`O``B``A`?Straight angle

AAcute angle

BReflex angle

CRight angle

D

In the diagram, $AC$`A``C` is a tangent to the circle with centre $O$`O`. What is the measure of $x$`x`? Give reasons for your answer.

Two tangents are drawn from an external point $B$`B` to the circle with centre $O$`O`. What is the value of angle $x$`x`? Give reasons for your answer.