South African Mathematics Olympiad First Round

Question

In RUG, RU = 10 cm, UG = 21 cm, RG = 17 cm and RY is perpendicular to UG at point Y. The length of RY, in cm, is FIGURE_0 (A) 8 (B) 5 (C) 7 (D) 6 (E) 8.1

Accepted Answer

Let h = RY and x = UY, so GY = 21 - x. By Pythagoras' theorem, h^2 = 10^2 - x^2 and also h^2 = 17^2 - (21-x)^2, so 100 - x^2 = 289 - 441 + 42x - x^2. This gives 42x = 100 - 289 + 441 = 252 and therefore x = 252 42 = 6. Finally, h^2 = 100 - 6^2 = 64, so h = 8.