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Equal tangents from an external point

Tangents drawn from the same external point to a circle are equal in length.

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Student-friendly explanation

If an external point P touches the circle at A and B, then PA = PB. This theorem is extremely important in proofs and length-finding questions. It is not enough to know that the tangents exist; you must also remember that their lengths are equal.

How to write this in exams

  1. 1

    Start with the exact idea

    Tangents drawn from the same external point to a circle are equal in length.

  2. 2

    Then show how to use it

    1. Identify the external point. 2. Check that both lines are tangents. 3. Use the theorem that tangents from the same external point are equal. 4. Substitute the equal lengths and solve.

  3. 3

    Add one concrete example

    From external point P, if PA and PB are tangents to the circle, then PA = PB.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to compare tangents from different points or to say only the arcs are equal. The theorem is about tangent lengths from the same outside point.

Definition

Tangents drawn from the same external point to a circle are equal in length.

Example

From external point P, if PA and PB are tangents to the circle, then PA = PB.

Rule to remember

If PA and PB are tangents from the same external point P, then PA = PB.

Memory hook

Same outside point, same tangent length.

Examples and method

Worked example

If PA = 7 cm and PB is another tangent from the same outside point P, then PB is also 7 cm.

Method to apply

1. Identify the external point. 2. Check that both lines are tangents. 3. Use the theorem that tangents from the same external point are equal. 4. Substitute the equal lengths and solve.

Diagram support

Draw the outside point, the two tangent segments, and label both contact points clearly.

How CBSE asks it

This theorem is often asked in proof steps and in finding an unknown tangent length.

Avoid common mistakes

Common confusion

Students sometimes treat tangent lengths from different external points as equal. The theorem works only for tangents from the same external point.

Common wrong answer

A common wrong answer is to compare tangents from different points or to say only the arcs are equal. The theorem is about tangent lengths from the same outside point.

Exam tip

Whenever two tangents start from one outside point, immediately write their lengths equal.

Quick check

Two tangents are drawn from the same point outside a circle. What can be said about their lengths?

Their lengths are equal. Tangents drawn from the same external point to a circle are always equal in length.

Answer writing and exam use

1-mark answer

Tangents drawn from the same external point to a circle are equal in length.

2-mark answer

Tangents drawn from the same external point to a circle are equal in length. If PA and PB are tangents from the same external point P, then PA = PB. From external point P, if PA and PB are tangents to the circle, then PA = PB.

3-mark answer

If an external point P touches the circle at A and B, then PA = PB. This theorem is extremely important in proofs and length-finding questions. It is not enough to know that the tangents exist; you must also remember that their lengths are equal. If PA and PB are tangents from the same external point P, then PA = PB. If PA = 7 cm and PB is another tangent from the same outside point P, then PB is also 7 cm. This theorem is often asked in proof steps and in finding an unknown tangent length. A common wrong answer is to compare tangents from different points or to say only the arcs are equal. The theorem is about tangent lengths from the same outside point.
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