HMMT 二月 2003 · 几何 · 第 2 题

HMMT February 2003 — Geometry — Problem 2

As shown, U and C are points on the sides of triangle M N H such that M U = s , U N = 6, N C = 20, CH = s , HM = 25. If triangle U N C and quadrilateral M U CH have equal areas, what is s ? N 6 20 U s M C s 25 H

解析

As shown, U and C are points on the sides of triangle M N H such that M U = s , U N = 6, N C = 20, CH = s , HM = 25. If triangle U N C and quadrilateral M U CH have equal areas, what is s ? N 6 20 U s M C s 25 H Solution: 4 Using brackets to denote areas, we have [ M CH ] = [ U N C ] + [ M U CH ] = 2[ U N C ]. On the other hand, triangles with equal altitudes have their areas in the same ratio as their bases, so [ M N H ] [ M N H ] [ M N C ] N H M N s + 20 s + 6 2 = = · = · = · . [ U N C ] [ M N C ] [ U N C ] N C U N 20 6 Clearing the denominator gives ( s + 20)( s + 6) = 240, and solving the quadratic gives s = 4 or − 30. Since s > 0, we must have s = 4. 1