HMMT 十一月 2017 · 冲刺赛 · 第 12 题

HMMT November 2017 — Guts Round — Problem 12

[ 4 ] Trapezoid ABCD , with bases AB and CD , has side lengths AB = 28, BC = 13, CD = 14, and DA = 15. Let diagonals AC and BD intersect at P , and let E and F be the midpoints of AP and BP , respectively. Find the area of quadrilateral CDEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2017, November 11, 2017 — GUTS ROUND Organization Team Team ID#

解析

[ 4 ] Trapezoid ABCD , with bases AB and CD , has side lengths AB = 28, BC = 13, CD = 14, and DA = 15. Let diagonals AC and BD intersect at P , and let E and F be the midpoints of AP and BP , respectively. Find the area of quadrilateral CDEF . Proposed by: Christopher Shao Answer: 112 1 Note that EF is a midline of triangle AP B , so EF is parallel to AB and EF = AB = 14 = CD . We 2 also have that EF is parallel to CD , and so CDEF is a parallelogram. From this, we have EP = P C CE 2 2 as well, so = . It follows that the height from C to EF is of the height from C to AB . We can CA 3 3 calculate that the height from C to AB is 12, so the height from C to EF is 8. Therefore CDEF is a parallelogram with base 14 and height 8, and its area is 14 · 8 = 112.