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Circles
Introduction
Tangent to a Circle
Practice: Tangent Perpendicularity
Number of Tangents from a Point on a Circle
Practice: Tangent Lengths and Angles
Summary
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Glossary

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CirclesTangent to a Circle

Reading time: ~25 min

Every few weeks a knife-sharpener pushes his cart down Aman's street. When he presses a knife against the spinning stone wheel, sparks fly off — not in curves, and not in every direction. Each spark shoots away in a straight line that just grazes the wheel at the point where the spark broke free.

That grazing line is a tangent. As you saw in the last section, a tangent to a circle is a line that meets the circle at exactly point.

Note

The word tangent comes from the Latin word tangere, which means "to touch". The Danish mathematician Thomas Fincke introduced it in 1583.

But does every point of a circle have a tangent, and could it have more than one? Two experiments settle the question.

Activity 1: The Turning Wire

Take a circular wire and attach a straight wire AB to it at a point P, so that AB can turn about P like the hand of a clock. In most positions AB crosses the circle twice: at P and at a second point.

Fig. 10.3 (i) is live. Read the missions, then drag the handle at A to turn the wire about P.

As AB turned towards the position A′B′, the second point (Q on one side, R on the other) slid closer and closer to P, and in the position A′B′ it landed on P. Turning from either side gave the very same line. So at a point of a circle there is tangent.

The tangent to a circle is a special case of the , when the two end points of its corresponding coincide.

Activity 2: Parallel Slides

Here is a second way to see the same thing. Draw a circle and a secant PQ. Now draw lines parallel to PQ, further and further away from it on both sides, and watch the chords they cut off.

Fig. 10.3 (ii) is live. Drag the handle to slide a line parallel to the secant PQ. Every whole-centimetre position you pass leaves a faint trace, so you can watch the chords shrink.

The chords became shorter and shorter until, on each side, the chord shrank to a single point. Those two lines, P′Q′ and P″Q″, are the tangents parallel to PQ, and every line beyond them misses the circle. So a secant has exactly tangents parallel to it.

Activity 2 tells the same story as Activity 1: a tangent is the secant whose chord has shrunk until its two end points .

The common point of a tangent and the circle is called the point of contact, and the tangent is said to touch the circle there. In Fig. 10.1 (iii), the point of contact is A.

The Wheel and the Road

Now look at a bicycle wheel rolling along a road. Its spokes run from the hub to the rim, so every spoke lies along a of the wheel. The road meets the wheel at just one point at a time, so the road is a to the circle of the wheel.

Fig. 10.4 is live. Drag the wheel along the road and keep an eye on the radius to the point of contact and the road.

Wherever the wheel stood, the radius through the point of contact met the road at . Is that special to wheels and roads, or is it true for every circle and every tangent? Let us prove it.

Tangent Perpendicular to the Radius

Theorem 10.1 : The tangent at any point of a circle is perpendicular to the radius through the point of contact.

First an experiment. In Fig. 10.5 the line XY is the tangent at P and OP is the radius through P. Drag the point Q along XY and watch the length of OQ.

Proof : We are given a circle with centre O and a tangent XY that touches it at P. We have to prove that OP \perp XY. Take any point Q on XY other than P, and join OQ.

Q cannot lie inside the circle: then XY would cut the circle at two points and be a secant, not a tangent. Q is not on the circle either, because P is the only point XY shares with it. So Q lies the circle, and therefore OQ is the radius OP:

OQ > OP

This is true for every point Q of XY except P itself. So OP is the of all the distances from O to the points of XY, and the shortest segment from a point to a line is the one (Theorem A1.7). Hence OP \perp XY.

Why the shortest segment is the perpendicular one

Drop the perpendicular from O to the line, and join O to any other point Q of the line. The two segments form a right triangle with the right angle at the foot of the perpendicular. OQ is the hypotenuse, the side opposite the right angle, so it is always longer than the perpendicular. That is why the shortest distance from O to XY is measured along the radius OP.

Remarks

  1. Through P there is exactly one line perpendicular to OP, so at any point of a circle there is tangent.
  2. The line through the point of contact that contains the radius is called the normal to the circle at that point.

Speed Round: Radius, Distance, Tangent

Theorem 10.1 hands you a right angle at the point of contact. So the radius OP, the tangent segment PQ and the line OQ to the centre always form a right triangle, with OQ as its hypotenuse:

OP^2 + PQ^2 = OQ^2

Find the missing length. Get five right to move on.

Sina