PQR is a right angled triangle in which the right angle is at Q and PQ = QR. If the hypotenuse PR is 20 cm, what is the area, in square centimetres, of triangle PQR?

Introduction / Context:
This problem is a classic example of a right isosceles triangle, where two legs are equal and the hypotenuse is given. Such questions are very common in aptitude exams and test familiarity with the relationship between the legs and the hypotenuse as well as the formula for the area of a right triangle.

Given Data / Assumptions:

• Triangle PQR is right angled at Q.
• PQ = QR, so it is an isosceles right triangle.
• The hypotenuse PR has length 20 cm.
• We are asked to find the area of triangle PQR in square centimetres.

Concept / Approach:
In a right isosceles triangle with legs of length a and hypotenuse h, the Pythagoras theorem gives: h^2 = a^2 + a^2 = 2a^2 so: a^2 = h^2 / 2 and: a = h / √2 Once we know the leg length a, the area of a right triangle is: Area = (1/2) * (leg 1) * (leg 2) = (1/2) * a * a = (1/2) * a^2

Step-by-Step Solution:
Step 1: Use the Pythagoras theorem for the isosceles right triangle. Given hypotenuse PR = 20 cm, let PQ = QR = a. Then h^2 = 2a^2, so 20^2 = 2a^2. 400 = 2a^2 a^2 = 400 / 2 = 200 Step 2: Compute the area using a^2. Area = (1/2) * a^2 = (1/2) * 200 = 100 square centimetres. Step 3: There is no need to compute a explicitly; using a^2 is enough to find the area.

Verification / Alternative check:
Alternatively, compute a explicitly: a = h / √2 = 20 / √2 = (20√2) / 2 = 10√2 Then: Area = (1/2) * a * a = (1/2) * (10√2) * (10√2) = (1/2) * 100 * 2 = 100 This agrees with the earlier calculation and confirms that the area is 100 square centimetres.

Why Other Options Are Wrong:
Option 2: 100√2 sq cm is larger than the correct area and would arise from incorrect manipulation of the square root or from neglecting the factor of 1/2 in the area formula. Option 3: 50 sq cm is exactly half the correct area, suggesting that someone may have used a^2 instead of (1/2) * a^2 or may have misapplied the Pythagoras theorem. Option 4: 50√2 sq cm again indicates confusion between the hypotenuse and the leg length or misapplication of square roots. Option 5: None of these is incorrect because 100 sq cm is present and is the correctly computed area.

Common Pitfalls:
A frequent error is to forget that the legs are equal in a right isosceles triangle, or to incorrectly set up the Pythagoras relation as h^2 = a^2 instead of h^2 = 2a^2. Another pitfall is forgetting the 1/2 factor in the triangle area formula, which leads to an answer that is double the correct value. Some learners also confuse the hypotenuse with one of the legs, especially when numbers are neat, such as 20. Writing out each equation carefully helps avoid these mistakes.

Final Answer:
The area of triangle PQR is 100 sq cm.