HMMT 十一月 2023 · 冲刺赛 · 第 3 题

HMMT November 2023 — Guts Round — Problem 3

[5] Let ABCD be a rectangle with AB = 20 and AD = 23. Let M be the midpoint of CD , and let X be the reflection of M across point A . Compute the area of triangle XBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2023, November 11, 2023 — GUTS ROUND Organization Team Team ID#

解析

[5] Let ABCD be a rectangle with AB = 20 and AD = 23. Let M be the midpoint of CD , and let X be the reflection of M across point A . Compute the area of triangle XBD . Proposed by: Daniel Xianzhe Hong Answer: 575 Solution: X B A C D M Observe that [ XBD ] = [ BAD ] + [ BAX ] + [ DAX ]. We will find the area of each of these triangles individually. 1 • We have [ ABD ] = [ ABCD ]. 2 • Because AM = AX , [ BAX ] = [ BAM ] as the triangles have the same base and height. Thus, as 1 [ BAM ] have the same base and height as ABCD , [ BAX ] = [ BAM ] = [ ABCD ]. 2 • From similar reasoning, we know that [ DAX ] = [ DAM ]. We have that DAM has the same base 1 and half the height of the rectangle. Thus, [ DAX ] = [ DAM ] = [ ABCD ]. 4 Hence, we have [ XBD ] = [ BAD ] + [ BAX ] + [ DAX ] 1 1 1 = [ ABCD ] + [ ABCD ] + [ ABCD ] 2 2 4 5 = [ ABCD ] 4 5 5 Thus, our answer is [ ABCD ] = (20 · 23) = 575. 4 4