HMMT 十一月 2017 · THM 赛 · 第 2 题

HMMT November 2017 — THM Round — Problem 2

H orizontal parallel segments AB = 10 and CD = 15 are the bases of trapezoid ABCD . Circle γ of radius 6 has center within the trapezoid and is tangent to sides AB , BC , and DA . If side CD cuts ◦ out an arc of γ measuring 120 , find the area of ABCD .

解析

H orizontal parallel segments AB = 10 and CD = 15 are the bases of trapezoid ABCD . Circle γ of radius 6 has center within the trapezoid and is tangent to sides AB , BC , and DA . If side CD cuts ◦ out an arc of γ measuring 120 , find the area of ABCD . Proposed by: Michael Tang 225 Answer: 2 ◦ Suppose that the center of the circle is O and the circle intersects CD at X and Y . Since ∠ XOY = 120 ◦ and triangle XOY is isosceles, the distance from O to XY is 6 · sin(30 ) = 3. On the other hand, the distance from O to AB is 6 as the circle is tangent to AB , and O is between AB and CD , so the height 9 · (10+15) 225 of the trapezoid is 6 + 3 = 9 and its area is = . 2 2