Tangents to a Circle

Rest a ball on a flat floor. It doesn't sink in and it doesn't hover — it touches the floor at exactly one point, right underneath its middle. Or picture a straight road running past a circular roundabout: for one instant the road runs alongside the roundabout, brushing its edge, without cutting across it. That grazing line is a tangent.

A tangent is a straight line that just touches a circle at a single point — the point of contact. It doesn't slice through (that would be a chord); it kisses the rim and carries on. And it comes with two lovely, genuinely useful properties. First: where it touches, the tangent makes a perfect right angle with the radius drawn to that point.

\text{tangent} \perp \text{radius at the point of contact}

Second: draw two tangents from the same outside point and something neat happens — they have equal length, so the two tangents, the two radii and the centre fold into a symmetric kite.

For a line touching a circle:

a tangent touches the circle at exactly one point;
the tangent is perpendicular to the radius at the point of contact — the angle there is 90^\circ;
the two tangents from one external point are equal in length;
so the external point, the two points of contact and the centre form a symmetric kite, with the two right angles at the contact points.

Why it works

Watch a single external point send two tangents to the circle. Step through it and notice the two right angles and the two equal lengths.

Because both right-angled triangles OT_1E and OT_2E share the side OE and have equal radii, they are mirror images — so the tangent lengths ET_1 and ET_2 match.

That right angle is a gift, because it turns a tangent problem into a Pythagoras problem. It is the same right-angle-hiding-in-a-circle theme you meet with the angle in a semicircle.

Worked examples

Example 1 — find a tangent length with Pythagoras. A circle has radius 5. From a point E that is 13 away from the centre O, a tangent is drawn. How long is it?

The radius, the tangent and the line OE form a right-angled triangle, with the right angle at the point of contact and OE as the hypotenuse. So: \text{tangent}^2 + 5^2 = 13^2 \;\Rightarrow\; \text{tangent}^2 = 169 - 25 = 144 \;\Rightarrow\; \text{tangent} = 12. The tangent from E is 12 long. (Notice the 5, 12, 13 right-angled triangle hiding inside.)

Example 2 — use "two tangents are equal" to solve for x. Two tangents are drawn from the same external point. One is labelled 3x - 2 and the other x + 6. Find x and the tangent length.

Two tangents from one point are equal, so set the lengths equal and solve: 3x - 2 = x + 6 \;\Rightarrow\; 2x = 8 \;\Rightarrow\; x = 4. Then each tangent is x + 6 = 4 + 6 = 10 long. (Check with the other expression: 3(4) - 2 = 10 — same answer, good.)

Example 3 — a belt around a pulley. A drive belt runs from a small wheel, straight across to a big wheel, wraps around, and comes straight back. Each straight run of the belt is a tangent to both wheels, and it leaves and rejoins each wheel at a right angle to that wheel's radius. That perpendicular is exactly why the belt runs dead straight between the wheels instead of sagging into the gap — the tangent line is the shortest, straightest bridge from one circle to the next. Every bicycle chain, every conveyor belt, every pulley system leans on this fact.

Two things trip people up:

The right angle only happens at the point of contact. The radius to the touch point meets the tangent at 90^\circ — but a radius drawn to any other point of the circle is not perpendicular to that tangent. If you are using the right angle, make sure you have drawn the radius to the exact point where the line touches.
"Two equal tangents" needs the same external point. The equal-length rule is about the two tangents drawn from one outside point. Two tangents drawn from different points have no reason to be equal — don't set their lengths equal by mistake.

The radius-meets-tangent-at-a-right-angle fact is why a ball, a coin or a bicycle wheel touches the ground at exactly one point, directly below its centre. Roll it, and that single touch point simply slides along — the wheel is always "tangent" to the road.

But the tangent idea did something far bigger. For centuries mathematicians asked: a circle has a clean tangent line at every point — could any curve have a tangent line, a straight line showing which way it is heading at each point? Answering "what is the slope of the tangent to a curve here?" is exactly the question Newton and Leibniz cracked in the 1600s, and their answer became the derivative — the founding idea of all of calculus. So this humble circle fact is a doorway straight into higher maths.