Hong Kong
Stage 4 - Stage 5

# Tangents to Circles

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.

## Multiple Tangents?

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

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 $\triangle OMP$OMP and $\triangle OQP$OQP:

$OM=OQ$OM=OQ(radii in a circle are equal)

$OP$OP is common

$\angle OMP=\angle OQP$OMP=OQP$=$=$90^\circ$90° (tangents meet radii at right angles)

$\therefore$ $\triangle OMP$OMP$\cong$$\triangle OQP$OQP (RHS)

$\therefore$ $MP=QP$MP=QP (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.

#### Worked Examples

##### Question 1

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.

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

$OB=OC$OB=OC

A

$OB>OC$OB>OC

B