HMMT 二月 2013 · 几何 · 第 9 题

HMMT February 2013 — Geometry — Problem 9

Pentagon ABCDE is given with the following conditions: ◦ ◦ (a) ∠ CBD + ∠ DAE = ∠ BAD = 45 , ∠ BCD + ∠ DEA = 300 √ √ √ BA 2 2 7 5 15 2 (b) = , CD = , and DE = DA 3 3 4 2 (c) AD · BC = AB · AE · BD Compute BD .

解析

Pentagon ABCDE is given with the following conditions: ◦ ◦ (a) ∠ CBD + ∠ DAE = ∠ BAD = 45 , ∠ BCD + ∠ DEA = 300 √ √ √ BA 2 2 7 5 15 2 (b) = , CD = , and DE = DA 3 3 4 2 (c) AD · BC = AB · AE · BD Compute BD . √ AD 3 √ Answer: 39 As a preliminary, we may compute that by the law of cosines, the ratio = . BD 5 AP Now, construct the point P in triangle ABD such that △ AP B ∼ △ AED . Observe that = AD AE · AB BC = (where we have used first the similarity and then condition 3). Furthermore, ∠ CBD = AD · AD BD ∠ DAB − ∠ DAE = ∠ DAB − ∠ P AB = ∠ P AD so by SAS, we have that △ CBD ∼ △ P AD . Geometry Test AB AD Therefore, by the similar triangles, we may compute P B = DE · = 5 and P D = CD · = 7. AD BD Furthermore, ∠ BP D = 360 − ∠ BP A − ∠ DP A = 360 − ∠ BCD − ∠ DEA = 60 and therefore, by the √ law of cosines, we have that BD = 39.