HMMT 二月 2013 · 几何 · 第 9 题
HMMT February 2013 — Geometry — Problem 9
题目详情
英文原题
Pentagon ABCDE is given with the following conditions: (a)∠CBD +∠DAE =∠BAD = 45◦,∠BCD +∠DEA = 300◦ (b)BA DA=2√ 2 3,CD =7√ 5 3, and DE =15√ 2 4 (c)AD2·BC=AB·AE·BD Compute BD.
解析
英文解析
- 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
(c) AD · BC = AB · AE · BD2
Compute BD .
√
AD 3
√
Answer: 39 As a preliminary, we may compute that by the law of cosines, the ratio = .
5BD
Now, construct the point P in triangle ABD such that △ AP B ∼ △ AED . Observe that =AP
AE · AB BCAD = (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.