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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete 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.

Definition

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

Example

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

Rule to remember

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

Memory hook

One outside point, equal tangent lengths.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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

2-mark answer

Two tangent segments drawn from the same external point are equal, and this fact is often used with segment labels in figures. From the same external point, tangent segments are equal in length. If P is outside the circle and PA and PB are tangents, then PA = PB.

3-mark answer

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. From the same external point, tangent segments are equal in length. If PA = x + 3 and PB = 9, then x + 3 = 9, so x = 6. The equality comes from the tangent theorem. This appears in algebraic geometry questions where side expressions are given on a circle figure. 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.
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