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Tangent length in a figure

Tangent lengths from the same external point are equal, which helps find unknown lengths in figures.

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Student-friendly explanation

In many drawings, one tangent length is given and the other is unknown. The quickest way is to identify that both tangents start from the same outside point. Then you can directly set the lengths equal and solve for the unknown.

How to write this in exams

  1. 1

    Start with the exact idea

    Tangent lengths from the same external point are equal, which helps find unknown lengths in figures.

  2. 2

    Then show how to use it

    1. Identify the external point. 2. Check that both segments are tangents. 3. Set the two tangent lengths equal. 4. Solve the equation for the unknown.

  3. 3

    Add one concrete example

    If PA and PB are tangents from P and PA = 8 cm, then PB = 8 cm.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to assume the two lengths are different because the drawing looks uneven. Geometry does not depend on visual size in the sketch.

Definition

Tangent lengths from the same external point are equal, which helps find unknown lengths in figures.

Example

If PA and PB are tangents from P and PA = 8 cm, then PB = 8 cm.

Rule to remember

Tangents from the same external point are equal: PA = PB.

Memory hook

Same start point, equal tangent lengths.

Examples and method

Worked example

If PA = 5x - 1 and PB = 3x + 7, then 5x - 1 = 3x + 7, so x = 4 and each tangent is 19 cm.

Method to apply

1. Identify the external point. 2. Check that both segments are tangents. 3. Set the two tangent lengths equal. 4. Solve the equation for the unknown.

Diagram support

Label the common external point and mark both tangent segments clearly.

How CBSE asks it

This is asked as a direct length-finding question or inside algebraic geometry problems.

Avoid common mistakes

Common confusion

Students sometimes add the tangent lengths instead of setting them equal.

Common wrong answer

A common wrong answer is to assume the two lengths are different because the drawing looks uneven. Geometry does not depend on visual size in the sketch.

Exam tip

If two tangent segments begin from the same outside point, write them equal before doing anything else.

Quick check

If two tangents from the same external point are 9 cm and unknown, what is the unknown length?

The unknown length is 9 cm. Tangents drawn from the same external point are always equal in length.

Answer writing and exam use

1-mark answer

Tangent lengths from the same external point are equal, which helps find unknown lengths in figures.

2-mark answer

Tangent lengths from the same external point are equal, which helps find unknown lengths in figures. Tangents from the same external point are equal: PA = PB. If PA and PB are tangents from P and PA = 8 cm, then PB = 8 cm.

3-mark answer

In many drawings, one tangent length is given and the other is unknown. The quickest way is to identify that both tangents start from the same outside point. Then you can directly set the lengths equal and solve for the unknown. Tangents from the same external point are equal: PA = PB. If PA = 5x - 1 and PB = 3x + 7, then 5x - 1 = 3x + 7, so x = 4 and each tangent is 19 cm. This is asked as a direct length-finding question or inside algebraic geometry problems. A common wrong answer is to assume the two lengths are different because the drawing looks uneven. Geometry does not depend on visual size in the sketch.
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