Solution for CAT Geometry Question 1

1. The unequal side of an isosceles triangle is thrice the inradius of the triangle. What is the ratio of the tangent side or the shortest side of the isosceles triangle Inradius = r = ID BC = 3r AB = AC = kr (say) Triangle ADC is right angled AD 2 + DC 2 = AC 2 AD 2 = AC 2 - DC 2 AD 2 = k 2 r 2 - Area = rs = r A I D B C

2. Area 3+2k = 4(4k 2 + 9 +12k) = 9(4k 2 - 9)

3. (3 +2k) 2 = 4k 2 - 9 4(4k 2 + 9 +12k) = 9(4k 2 - 9) 16k 2 + 36 + 48k = 36k 2 - 81 20k 2 - 48k - 117 = 0 20k 2 + 36k - 78k - 117 = 0 10k(2k + 3) - 39(2k + 3) = 0 (10k - 39)(2k + 3) = 0 K = ; Ratio =