jmc

Question

In convex quadrilateral ABCD, A C, AB = CD = 180, and AD BC. The perimeter of ABCD is 640. Find 1000 A . (The notation x means the greatest integer that is less than or equal to x.)

Accepted Answer

FIGURE By the Law of Cosines on ABD at angle A and on BCD at angle C (note C = A), 180^2 + AD^2 - 360 AD A = 180^2 + BC^2 - 360 BC A(AD^2 - BC^2) = 360(AD - BC) A(AD - BC)(AD + BC) = 360(AD - BC) A(AD + BC) = 360 AWe know that AD + BC = 640 - 360 = 280. A = 280360 = 79 = 0.777 1000 A = 777.