Course 12.3Lesson 1 of 3

What a tangent is

A line that touches a circle at exactly one point is a tangent. The key fact is that a tangent is at right angles to the radius drawn to the point where it touches. From this one theorem we can find lengths and angles that look hard at first.

about 30 min1 simulation1 practice question1 Python program5 questions to check yourself

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Try the code and the simulations beside the lesson.

A wheel on the road

Look at a bicycle wheel standing on a flat road. The wheel and the road touch at one point. Everywhere else the road stays below the wheel. A straight line that touches a circle in this way is called a tangent, and the point where it touches is the point of contact.

Not every line is like this. A straight line and a circle can meet in three different ways.

A line that meets the circle at two points is a secant. The part of it inside the circle is a chord.
A line that meets the circle at exactly one point is a tangent.
A line that does not meet the circle at all misses it.

Two pictures side by side. On the left, a circle with centre O sits on a horizontal green line that touches it at one point P. The radius OP is drawn straight down and a small square marks the right angle between OP and the line. A dashed grey line from O to another point Q on the green line is longer than OP, and a note says so. On the right, the same circle and the point P at its bottom, with three red lines through P that cut the circle again at three points getting closer and closer to P. The flattest red line is almost the green line, and the caption says that the secant becomes the tangent when the second point slides onto P.

How the distance decides

Let the circle have centre O and radius $r$ . Let a line be at a distance $d$ from O. The distance from a point to a line is the length of the perpendicular from the point to the line. The line meets the circle in these ways:

If $d < r$ , the line cuts the circle at two points. It is a secant.
If $d = r$ , the line touches the circle at one point. It is a tangent.
If $d > r$ , the line is outside the circle. It misses it.

So a tangent is the special line that is exactly one radius away from the centre. Try it in the simulation at the end of this lesson.

A secant that turns into a tangent

Fix a point P on a circle. Draw a secant through P that cuts the circle again at a point Q. Now slide Q along the circle towards P. The secant turns about P. When Q reaches P, the two points of the chord have become one, and the secant has become the tangent at P.

This tells us two things. A tangent is the limit of a secant. And at each point of a circle there is exactly one tangent.

The tangent and the radius

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

Proof. Let O be the centre and let the tangent XY touch the circle at P. We must show that $OP \perp XY$ .

Take another point. Let Q be any point on XY other than P.
Q is outside. The tangent meets the circle only at P. So Q lies outside the circle, and $OQ > OP$ .
Every other point is farther. The same is true for every point Q on XY except P. So P is the point of XY that is nearest to O.
The shortest distance is perpendicular. The shortest distance from a point to a line is the perpendicular. So $OP \perp XY$ . $\blacksquare$

The converse is also true. A line drawn through the end of a radius, at right angles to it, is a tangent. Every other point of such a line is farther from O than P is, so it lies outside the circle.

Using the theorem

Whenever you see a tangent, draw the radius to the point of contact and mark the right angle. This makes a right triangle.

A length. A tangent at P is drawn to a circle of radius $6$ cm with centre O. T is a point on the tangent with $OT = 10$ cm. Find $PT$ .

Draw the right triangle. $OP \perp PT$ , so $\triangle OPT$ has a right angle at P.
Use Pythagoras. $PT^2 = OT^2 - OP^2 = 100 - 36 = 64$ .
Take the root. $PT = 8$ cm.

An angle. In the same figure, $\angle OTP = 28^\circ$ . Find $\angle POT$ .

The triangle has a right angle at P. The three angles add up to $180^\circ$ .
Subtract. $\angle POT = 180^\circ - 90^\circ - 28^\circ = 62^\circ$ .

How many tangents from a point?

From a point inside the circle, every line through the point is a secant. So no tangent can be drawn.
From a point on the circle, there is exactly one tangent, the one at that point.
From a point outside the circle, there are exactly two tangents. We study them in the next lesson.

Practise with new numbers

Try it in Python

The program moves a line across a circle of radius $5$ cm. For each distance it says whether the line is a secant, a tangent or outside the circle.

Common mistakes

Thinking a tangent must be short. A tangent is a whole line. It touches the circle at one point and continues for ever.
Forgetting the right angle. The tangent is perpendicular to the radius, not to the chord.
Mixing up secant and tangent. A secant cuts the circle at two points. A tangent touches it at one.
Using the wrong distance. The distance of a line from the centre is the perpendicular distance.

Remember these

A tangent meets a circle at exactly one point.
The distance from the centre to a tangent equals the radius.
The tangent at a point is perpendicular to the radius through that point.
From an outside point, two tangents can be drawn. From a point on the circle, one. From a point inside, none.