medium difficulty5 min

Proof using congruent triangles

Many tangent results are proved by showing two triangles are congruent.

In one line: Equal parts first, congruence next, result after that.

Understand it

Circle proofs often become simple once you spot two triangles with equal sides and a common side or right angle. In tangents, the radii to the points of contact are equal, the tangent-radii angles are right angles, and this often helps prove triangle congruence. Then equal tangent lengths or other results follow.

Example

In triangles OAP and OBP, if OA = OB and OP is common, triangle congruence can help prove PA = PB.

Formula and rule

Use congruence tests such as RHS, SAS, or SSS when the required equal parts are available.

See it worked out

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

Worked example

If OA = OB and OP is common, and both radii meet tangents at right angles, then triangles OAP and OBP can be proved congruent, giving PA = PB.

Method you can reuse

1Identify the two triangles.
2Mark equal radii and right angles.
3Choose the correct congruence test.
4Write the congruence statement.
5Use CPCT to get the required result.

Sketch it in your answer

Mark equal radii, the common side, and right angles before writing the proof.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is triangle congruence often used in tangent proofs?

Think first, then show answer

Because the radii to the points of contact are equal and the tangent makes a right angle, so two triangles often match in enough parts to prove them congruent.

Common wrong answer

A common wrong answer is to stop after writing that two radii are equal. Equal radii alone do not complete the proof.

Don't mix this up

Students sometimes try to prove tangent results by comparing only one side or one angle. A full congruence argument needs enough matching parts.

Exam tip

Look for equal radii, common side, and right angles. These are the usual clues for triangle congruence in circle proofs.

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

Long-answer questions often ask you to prove equal tangents or a perpendicular relation using triangle congruence.

1
Enough for 1 mark
Many tangent results are proved by showing two triangles are congruent.
2
Add for 2 marks
Example: In triangles OAP and OBP, if OA = OB and OP is common, triangle congruence can help prove PA = PB.
3
Add for 3 marks
In tangents, the radii to the points of contact are equal, the tangent-radii angles are right angles, and this often helps prove triangle congruence.

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