2023 · Q3
The right-angled triangle ABC is such that ∠B = 90°. Point D is picked on BC such that triangles ABC and DBA are similar. If AB : BC = m : n, what is ΔABC : ΔABD, where Δ denotes the area of a triangle?
general-mental-abilityquantitative-aptitude
- An : m
- Bn² : m²
- C(m + n) : n
- D(m + n)² : n²
Explanation
Write AB = m and BC = n. From the similarity of triangles ABC and DBA, the sides about the right angle are proportional: AB/DB = BC/AB, giving DB = AB^2/BC = m^2/n. The linear scale factor from triangle DBA to triangle ABC is therefore AB/DB = n/m. Since areas of similar triangles are in the duplicate (square) ratio of corresponding sides, area(ABC) : area(ABD) = (n/m)^2 = n^2 : m^2. Hence option B.