In right triangle PQR with ∠PQR = 90°, the leg PQ = 10 cm and the hypotenuse PR = 26 cm. Find the inradius r (radius of the incircle) in centimetres.

Introduction / Context:The inradius r of a right triangle has a convenient formula in terms of side lengths: if the legs are a and b and the hypotenuse is c, then r = (a + b − c) / 2. We will first find the missing leg using the Pythagorean theorem, then apply the inradius formula.

Given Data / Assumptions:

• Right angle at Q, so PR is the hypotenuse.
• PQ = 10 cm, PR = 26 cm.

Concept / Approach:Use Pythagoras to get the other leg QR. For a right triangle, r = (a + b − c)/2, where c is the hypotenuse.

Step-by-Step Solution:

Verification / Alternative check:Area K = (1/2) * a * b = (1/2) * 10 * 24 = 120. Also, K = r * s with semiperimeter s = (a + b + c)/2 = (10 + 24 + 26)/2 = 30. Then r = K/s = 120/30 = 4, consistent.

Why Other Options Are Wrong:

• 9, 8, 6: Do not satisfy r = (a + b − c)/2 for the given triple 10–24–26.

Common Pitfalls:Using (a + b + c)/2 instead of (a + b − c)/2; mixing up which side is the hypotenuse.

Final Answer:4