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 △ABC : △ABD, where △ denotes the area of a triangle?

In triangle ABC, ∠B = 90° and AB:BC = m:n. Given triangle ABC is similar to triangle DBA, we identify corresponding sides. In similar triangles, the ratio of areas is equal to the square of the ratio of their corresponding sides. For △ABC ~ △DBA, the hypotenuse of △ABC is AC and the hypotenuse of △DBA is AB. Using the Pythagorean theorem, AC² = AB² + BC² = m² + n². The ratio of the area of △ABC to △ABD is (AC/AB)², which equals (m² + n²)/m². However, the question asks for the ratio based on the similarity mapping where AB:BC = m:n. If △ABC ~ △DBA, then AB/DB = BC/BA = AC/DA. From BC/BA = n/m, we find DB = AB * (AB/BC) = m * (m/n) = m²/n. The ratio of areas of triangles with the same height (AB) is the ratio of their bases: BC/BD = n / (m²/n) = n² / m². Adding the segments for the full ratio leads to (m + n)²: n² based on the geometric configuration of similar right triangles.