JEE Advanced Maths: Triangles (Geometry) questions with solutions

Q1. In right triangle ABC, the right angle is at B, and BD is the altitude from B to hypotenuse AC. Given BD = 9 cm, and the inradii of triangles ABC, ABD, and BCD are r1, r2, and r3 respectively, find r1 + r2 + r3.

1. 9
2. 18
3. 27
4. 6

Answer: 9

There is a beautiful theorem: if BD is the altitude from the right angle B to hypotenuse AC in right triangle ABC, then r_ABC + r_ABD + r_BCD = BD. This follows because for a right triangle with legs p, q and hypotenuse h, the inradius is (p + q - h)/2. Letting AD = m, DC = n, BD = h, AB = sqrt(mh_total... applying the identity carefully, the sum telescopes to BD = 9.

Q2. In triangle ABC, points D and E lie on sides AB and AC respectively with DE parallel to BC. Given AD = 4x - 3, AE = 8x - 7, BD = 3x - 1, and CE = 5x - 3, find x.

1. x = 1
2. x = 2
3. x = 3
4. x = 4

Answer: x = 1

Since DE is parallel to BC, the Basic Proportionality Theorem gives AD/DB = AE/EC. Cross-multiplying and simplifying produces a quadratic; the valid value (giving positive lengths) is x = 1.