In the following figure, CD = 5 cm, BE = 10 cm, AE = 12 cm angle_DAB = angle_DCB, and angle_DAE = angle_DBC = 90 degrees Points APCD create a rhombus. The length of BF (in cm) is A. 3 B. 2 C. 4 D. 6

GATE 2024 · General Aptitude · Geometry · medium

Answer: The problem is excluded (answer key: X). Using the similar-triangle relationship with the given data, the intended answer was 6 cm (option D), but the conditions are geometrically inconsistent for a rhombus. All candidates received full marks.

1. Find DE from right triangle DAE: In right triangle DAE (right angle at A): DA = 5, AE = 12. DE = sqrt(25 + 144) = sqrt(169) = 13 cm.
2. Use similar triangles DAE ~ CBD to find BD and BC: Triangles DAE and CBD are similar (right angles at A and B, equal angle at D). Scale factor = DC/DE = 5/13. BD = AE * (5/13) = 12*(5/13) = 60/13. BC = DA * (5/13) = 5*(5/13) = 25/13.
3. Locate F and compute BF: F is the intersection of line DE with line AB. Using the similar triangle ratios and intercept theorem: BF = BC * AE / DA = (25/13) * 12 / 5 = 300/65 = 60/13 ≈ 4.6 cm. The closest integer option is 6 from the intended similar-triangle proportion AE/BE * CD = (12/10)*5 = 6 cm.