JEE Advanced Maths: Properties of Triangles / Geometry questions with solutions

Q1. In triangle ABC, point D lies on CB with CD = 15 and DB = 9. AD bisects angle A, and angle ABC = 90°. Find the length AB.

1. 32
2. 18
3. 7
4. 24

Answer: 18

By the angle bisector theorem, AB/AC = BD/DC = 9/15 = 3/5, so let AB = 3k, AC = 5k. Triangle ABC is right-angled at B with BC = CD + DB = 24. Then AC² = AB² + BC²: (5k)² = (3k)² + 24² ⇒ 25k² - 9k² = 576 ⇒ 16k² = 576 ⇒ k² = 36 ⇒ k = 6. So AB = 3k = 18.