21.3 Tangents and chords — Learn

Prerequisites: identify a radius, chord and tangent; use right angles and isosceles triangles.

Objectives: apply the tangent-radius, equal-tangent and perpendicular-bisector chord facts, and the alternate segment theorem.

Tangents and radii

A radius drawn to the point where a tangent touches the circle is perpendicular to the tangent. It makes a right angle of $90^\circ$.

Example 1. A radius meets a tangent at $T$. One angle at $T$ is $90^\circ$. Find the adjacent angle on the straight tangent line.

Angles on a straight line total $180^\circ$.

Subtract the right angle: $180^\circ-90^\circ=90^\circ$.

Tangents drawn from the same point outside a circle have equal lengths.

Example 2. Two tangents from point $P$ touch a circle at $A$ and $B$. If $PA=5.4$ cm, find $PB$.

Tangents from the same external point are equal.

Therefore $PB=PA=5.4$ cm.

Chords and the alternate segment

A perpendicular from the centre to a chord bisects the chord. Bisects means cuts into two equal parts.

Example 3. A perpendicular from the centre meets a chord of length $12$ cm. Find each part of the chord.

The perpendicular bisects the chord.

Divide the length by two: $12\div2=6$ cm.

The alternate segment theorem says the angle between a tangent and a chord equals the angle in the opposite segment of the circle.

Example 4. The angle between a tangent and a chord is $47^\circ$. Find the angle in the alternate segment.

The alternate segment theorem gives equal angles.

So the angle is $47^\circ$.

Watch out. The perpendicular from the centre bisects a chord only when it meets the chord at $90^\circ$. Check for the right-angle mark or a stated perpendicular.

Progress check

The chapter problem says a radius meets the chord at right angles. What does this tell you about the two pieces of the chord? Use the result in the review.

You should now be able to

• use the right angle between a tangent and radius;
• show that tangents from one external point have equal lengths;
• use a perpendicular from the centre to bisect a chord;
• apply the alternate segment theorem.