medium difficulty5 min

Tangent length in a figure

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

In one line: Same start point, equal tangent lengths.

Understand it

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.

Example

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

Formula and rule

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

See it worked out

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

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 you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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

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.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

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

1
Enough for 1 mark
Tangent lengths from the same external point are equal, which helps find unknown lengths in figures.
2
Add for 2 marks
Example: If PA and PB are tangents from P and PA = 8 cm, then PB = 8 cm.
3
Add for 3 marks
In many drawings, one tangent length is given and the other is unknown.

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