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Tangent to a circle

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete 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.

Definition

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

Example

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

Rule to remember

A tangent touches a circle at exactly one point.

Memory hook

Tangent means touch once.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

Students may be asked to identify a tangent from a figure or to name the point of contact.

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Study the tangent to a circle diagram carefully

Use the labelled diagram to keep tangent to a circle clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of tangent to a circle in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on tangent to a circle.

Revision cue

Revise tangent to a circle through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

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

3-mark answer

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. A tangent touches a circle at exactly one point. A line l touches a circle at P and does not enter the circle. Therefore l is a tangent to the circle at P. Students may be asked to identify a tangent from a figure or to name the point of contact. 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.
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