HMMT 二月 2004 · GEN1 赛 · 第 7 题

HMMT February 2004 — GEN1 Round — Problem 7

Let ACE be a triangle with a point B on segment AC and a point D on segment CE such that BD is parallel to AE . A point Y is chosen on segment AE , and segment CY is drawn. Let X be the intersection of CY and BD . If CX = 5, XY = 3, what is the ratio of the area of trapezoid ABDE to the area of triangle BCD ? C B X D A Y E 1

解析

Let ACE be a triangle with a point B on segment AC and a point D on segment CE such that BD is parallel to AE . A point Y is chosen on segment AE , and segment CY is drawn. Let X be the intersection of CY and BD . If CX = 5, XY = 3, what is the ratio of the area of trapezoid ABDE to the area of triangle BCD ? Solution: 39 / 25 C B K X D A L Y E Draw the altitude from C to AE , intersecting line BD at K and line AE at L . Then CK is the altitude of triangle BCD , so triangles CKX and CLY are similar. Since CY /CX = 8 / 5, CL/CK = 8 / 5. Also triangles CKB and CLA are similar, so that CA/CB = 8 / 5, and triangles BCD and ACE are similar, so that AE/BD = 8 / 5. The area of ACE is (1 / 2)( AE )( CL ), and the area of BCD is (1 / 2)( BD )( CK ), so the 2 ratio of the area of ACE to the area of BCD is 64 / 25. Therefore, the ratio of the area of ABDE to the area of BCD is 39 / 25.