HMMT 十一月 2012 · GEN 赛 · 第 3 题

HMMT November 2012 — GEN Round — Problem 3

[ 3 ] ABCD is a rectangle with AB = 20 and BC = 3. A circle with radius 5, centered at the midpoint of DC , meets the rectangle at four points: W , X , Y , and Z . Find the area of quadrilateral W XY Z .

解析

[ 3 ] ABCD is a rectangle with AB = 20 and BC = 3. A circle with radius 5, centered at the midpoint of DC , meets the rectangle at four points: W , X , Y , and Z . Find the area of quadrilateral W XY Z . Answer: 27 Suppose that X and Y are located on AB with X closer to A than B . Let O be the center of the circle, and let P be the midpoint of AB . We have OP ⊥ AB so OP X and OP Y are right triangles with right angles at P . Because OX = OY = 5 and OP = 3, we have XP = P Y = 4 by the Pythagorean theorem. Now, W XY Z is a trapezoid with W Z = W O + OZ = 5 + 5 = 10 , ( ) 10+8 XY = XP + P Y = 8, and height 3, so its area is × 3 = 27. 2