Two triangles ΔABC and ΔPQR are similar. The area of triangle ΔABC is 121 square centimetres and the area of triangle ΔPQR is 64 square centimetres. If the side PQ of the smaller triangle is 12 cm, what is the length, in centimetres, of side AB of the larger triangle?

Introduction / Context: This question uses properties of similar triangles, which are common in geometry based aptitude problems. When two triangles are similar, the ratio of their areas is equal to the square of the ratio of corresponding sides. Here we must use the area ratio to find an unknown side length of the larger triangle. Given Data / Assumptions:

• ΔABC and ΔPQR are similar.
• Area of ΔABC = 121 cm^2.
• Area of ΔPQR = 64 cm^2.
• PQ corresponds to AB and PQ = 12 cm.
• Find AB in centimetres.

Concept / Approach: For similar triangles, if k is the ratio of a side in ΔABC to the corresponding side in ΔPQR, then the area ratio is k^2. So area(ABC) / area(PQR) = (AB / PQ)^2. We can use this relation to find AB since the area ratio and PQ are known. Step-by-Step Solution: Compute the area ratio: area(ABC) / area(PQR) = 121 / 64. Let AB / PQ = r. Then r^2 = 121 / 64. Take square roots: r = √(121 / 64) = 11 / 8. So AB / PQ = 11 / 8, and PQ = 12 cm. Therefore AB = (11 / 8) * 12 = 11 * 12 / 8. Simplify: 12 / 8 = 3 / 2, so AB = 11 * (3 / 2) = 33 / 2 = 16.5 cm. Verification / Alternative check: We can check the logic by squaring the side ratio again. AB / PQ = 16.5 / 12 = 33 / 24 = 11 / 8, so the area ratio becomes (11 / 8)^2 = 121 / 64, which matches the given data. This confirms that AB = 16.5 cm is consistent. Why Other Options Are Wrong: Options a, b, and d are significantly smaller and would produce an area ratio different from 121 / 64 when squared. Option e (18.5) would yield a side ratio greater than 11 / 8, which again does not match the squared area ratio. Only 16.5 gives the correct similarity ratio. Common Pitfalls: One frequent mistake is to assume the area ratio is directly equal to the side ratio, instead of its square. Another pitfall is to confuse which triangle is larger. Always compare the numerical areas first and confirm which triangle has the bigger area and therefore the larger sides. Final Answer: The length of side AB of the larger triangle is 16.5 cm.