Chapter 10 · Introduction

Chapter 10 — Circles

A cycle wheel touches the road at a single point; a stone whirled on a string flies off in a straight line when the string is let go. Meet the tangent — the line that just touches a circle — and the two surprising facts about it that solve a whole family of problems.

Watch a cycle wheel on a level road. However big the wheel is, it touches the road at just one point, and that point keeps moving as the wheel rolls. Now picture a stone tied to a string and whirled round your head. The moment you let go, the stone does not keep circling and it does not fly straight outwards along the string: it shoots off in a straight line that just grazes the circle it was following. Belts that drive a machine, a ruler laid against a coin, the rail of a curved track — in each case a straight line meets a circle at exactly one point. Such a line has a name: a tangent.

In this chapter you study tangents carefully. In Section 10.1 you meet the three ways a line can sit relative to a circle (it can cut it, touch it or miss it) and you prove that a tangent is always perpendicular to the radius drawn to the point where it touches. In Section 10.2 you stand at a point outside a circle, draw the two tangents from it, and prove that they have equal lengths. Together these two theorems are the whole toolkit: with Pythagoras' theorem and the angle facts you already know, they let you find lengths and angles in figures with circles, tangents and even quadrilaterals drawn around circles.

Secant: cuts a circle at two points→Tangent: touches it at one point→Tangent ⟂ radius at the point of contact→Two tangents from an outside point are equal→Quadrilateral around a circle: $AB+CD=BC+DA$

🎯 By the end of this chapter you can…

tell a secant from a tangent, name the point of contact, and decide from the perpendicular distance of a line from the centre whether the line cuts, touches or misses a circle;
state and prove that the tangent at a point of a circle is perpendicular to the radius through that point, and use it to find lengths with Pythagoras' theorem and angles with the angle sum of a triangle;
say how many tangents can be drawn to a circle from a point inside it, on it and outside it;
state and prove that the lengths of the two tangents from an external point are equal, and use it for lengths, angles and perimeters;
prove and use the fact that the opposite sides of a quadrilateral drawn around a circle have equal sums, including problems with an unknown $x$.

A stone moves round the circle on a string $OP$. Use the slider (or press Play) to choose where it is released. The red line is the straight path it takes: it touches the circle at $P$ and nowhere else. Check the first readout: however you move $P$, the angle between the string $OP$ and the red path is exactly $90^\circ$. That is the first big theorem of this chapter.

Release point $P$ (angle round the circle)40°

angle between $OP$ and the path—

radius $OP$ (cm)—

common points of path and circle—

distance of the path from $O$ (cm)—

Sections in this chapter

10.1

Tangent to a Circle

Secants and tangents, the point of contact, the three positions of a line relative to a circle, and the proof that a tangent is perpendicular to the radius at the point of contact.

tangent $\perp$ radius at the point of contact

Open section →10.2

Tangents from a Point Outside a Circle

The length of a tangent, the proof that the two tangents from an outside point are equal, and applications to angles, perimeters and quadrilaterals around a circle.

$PA=PB=\sqrt{OP^2-r^2}$

Open section →

✅ Before you start

You need, from Class 9: the parts of a circle (centre, radius, diameter, chord), the fact that the perpendicular from a point to a line is the shortest distance from the point to the line, congruent triangles (including the RHS criterion for right-angled triangles), the angle sum of a triangle ($180^\circ$) and of a quadrilateral ($360^\circ$), and Pythagoras' theorem. From this course you will use equations of the type $x^2+9^2=(x+3)^2$, which reduce to linear equations, and occasionally a quadratic (Section 4.2). Warm up: a right-angled triangle has legs $5$ cm and $12$ cm; find its hypotenuse (answer: $13$ cm).

← 9.2 Heights and Distances 10.1 Tangent to a Circle →