medium difficulty5 min

Tangent to a circle

A tangent to a circle is a line that touches the circle at exactly one point.

In one line: Tangent means touch once.

Understand it

A tangent does not cut the circle at two points like a secant. It just touches the circle at one point, called the point of contact. In exam questions, the first job is to identify whether the line is only touching or actually passing through the circle.

Example

If a line touches a circle at point A and no other point, then that line is a tangent at A.

Formula and rule

A tangent touches a circle at exactly one point.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1A line l touches a circle at P and does not enter the circle.
2Therefore l is a tangent to the circle at P.

Method you can reuse

1Look at the line and the circle.
2Count how many points are common.
3If there is only one common point, the line is a tangent.
4Name the point of contact clearly in the answer.

Sketch it in your answer

Mark the point where the line touches the circle and label it as the point of contact.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A line touches a circle at exactly one point. What is that line called?

Think first, then show answer

That line is called a tangent. It touches the circle at only one point, so it is not a secant or a chord.

Common wrong answer

A common wrong answer is to call a line a tangent just because it looks close to the circle. That is wrong unless it touches exactly once.

Don't mix this up

Students often call any line near a circle a tangent, even when it does not touch the circle at exactly one point.

Exam tip

Always count the contact points first. One touching point means tangent; two intersection points mean secant.