15th Irish 2002 Problem A1 The triangle ABC has a, b, c = 29, 21, 20 respectively. The points D, E lie on the segment BC with BD = 8, DE = 12, EC = 9. Find ∠DAE.
45 o Solution Put AD = x, AE = y. By repeated use of the cosine rule we have, cos B = (29 2 + 20 2 - 21 2 )/(2 · 20 · 29) = 20/29, x 2 = 20 2 + 8 2 - 2 · 8 · 20 · cos B = 7056/29, cos C = (21 2 + 29 2 - 20 2 )/(2 · 21 · 29) = 21/29, y 2 = 21 2 + 9 2 - 2 · 21 · 9 · cos C = 7200/29. Finally, cos DAE = (x 2 + y 2 - 12 2 )/(2 · x · y) = 1/√2. So angle DAE = 45 o . 15th Irish 2002 © John Scholes jscholes@kalva.demon.co.uk 15 June 2003 Last updated/corrected 15 Jun 03