In triangle PQR, angle PQR is a right angle (90°). The lengths of sides PQ and PR are 10 cm and 26 cm respectively. What is the length, in centimetres, of the inradius of the circle inscribed inside triangle PQR?

Introduction / Context:
This question deals with a right angled triangle and asks for the radius of its incircle, called the inradius. The incircle is the largest circle that can fit entirely inside the triangle and touch all three sides. Right triangle inradius formulas are very useful in geometry and competitive exams, particularly when combined with Pythagoras theorem.

Given Data / Assumptions:

• Triangle PQR is right angled at Q, so angle PQR = 90°.

• PQ = 10 cm.

• PR = 26 cm.

• Let QR be the third side, which we will find using Pythagoras theorem.

• We need the radius r of the incircle of triangle PQR.

Concept / Approach:
In a right angled triangle with legs a and b and hypotenuse c, the inradius r can be found using the formula r = (a + b − c) / 2. First we identify which side is the hypotenuse. Since the right angle is at Q, the side opposite to Q, which is PR, is the hypotenuse. Then we use Pythagoras theorem to compute the third side and finally apply the inradius formula.

Step-by-Step Solution:
Given angle at Q is 90°, so PR is the hypotenuse. Let PQ = a = 10 cm. Let QR = b (unknown) and PR = c = 26 cm. By Pythagoras theorem: a^2 + b^2 = c^2. So, 10^2 + b^2 = 26^2. 100 + b^2 = 676. b^2 = 676 − 100 = 576. b = QR = 24 cm. Now use inradius formula for a right triangle: r = (a + b − c) / 2. Substitute values: r = (10 + 24 − 26) / 2. r = (34 − 26) / 2 = 8 / 2 = 4 cm.

Verification / Alternative check:
Alternatively, we can use the fact that the area of the triangle can be expressed both as (1 / 2) * a * b and also as r * s, where s is the semi perimeter. First compute the area: Area = (1 / 2) * 10 * 24 = 120 square centimetres. Next compute the semi perimeter s = (a + b + c) / 2 = (10 + 24 + 26) / 2 = 60 / 2 = 30. Then r = Area / s = 120 / 30 = 4 cm. This matches the earlier inradius formula result.

Why Other Options Are Wrong:
The values 6, 8 and 9 are larger radii that would imply a larger area for the same semi perimeter or a different set of side lengths. They do not satisfy both the Pythagoras relationship and the inradius formula simultaneously. Only 4 cm is consistent with the true side lengths 10 cm, 24 cm and 26 cm and with the geometry of the incircle.

Common Pitfalls:
Errors often occur when identifying the hypotenuse or when applying Pythagoras theorem. Some students may incorrectly treat 10 cm as the hypotenuse or may forget to subtract the hypotenuse when using the inradius formula. Another pitfall is mixing up formulas for inradius and circumradius. Always verify that the numbers satisfy Pythagoras theorem before moving on to derived formulas like r = (a + b − c) / 2.

Final Answer:
The radius of the incircle of triangle PQR is 4 cm.