medium difficulty5 min

External point and tangent segments

Two tangent segments drawn from the same external point are equal, and this fact is often used with segment labels in figures.

In one line: One outside point, equal tangent lengths.

Practice This Concept

Understand it

In many problems, an outside point sends two line segments to the circle. Once both are confirmed as tangents, the equal-length rule can be used immediately. This helps in finding unknowns, proving equal sides, and solving geometry equations inside the figure.

Example

If P is outside the circle and PA and PB are tangents, then PA = PB.

Formula and rule

From the same external point, tangent segments are equal in length.

See it worked out

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

Worked example

1If PA = x + 3 and PB = 9, then x + 3 = 9, so x = 6.
2The equality comes from the tangent theorem.

Method you can reuse

1Confirm both segments are tangents.
2Check that they start from the same external point.
3Set the lengths equal.
4Solve for the unknown or complete the proof.

Sketch it in your answer

Label the outside point once and mark both tangent segments with equal signs if needed.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If two tangent segments come from the same external point, what relation holds between them?

Think first, then show answer

They are equal in length. This is one of the most useful circle properties for finding unknown side lengths in figures.

Common wrong answer

A common wrong answer is to think the lengths depend on which side of the circle they touch. They do not; the common external point decides equality.

Don't mix this up

Students sometimes compare the segments from the external point to two different contact points without checking whether both are tangents.

Exam tip

The moment you see two tangents from one outside point, write them equal and move to the algebra or proof step.

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 appears in algebraic geometry questions where side expressions are given on a circle figure.

1
Enough for 1 mark
Two tangent segments drawn from the same external point are equal, and this fact is often used with segment labels in figures.
2
Add for 2 marks
Example: If P is outside the circle and PA and PB are tangents, then PA = PB.
3
Add for 3 marks
In many problems, an outside point sends two line segments to the circle.

MCQ Quiz

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