HMMT 十一月 2011 · 冲刺赛 · 第 18 题
HMMT November 2011 — Guts Round — Problem 18
题目详情
英文原题
- [ 10 ] Consider a cube ABCDEF GH , where ABCD and EF GH are faces, and segments AE, BF, CG, DHare edges of the cube. Let P be the center of face EF GH , and let O be the center of the cube. Giventhat AG = 1, determine the area of triangle AOP .
4 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUNDTH
Round 7
解析
英文解析
- [ 10 ] Consider a cube ABCDEF GH , where ABCD and EF GH are faces, and segments AE, BF, CG, DHare edges of the cube. Let P be the center of face EF GH , and let O be the center of the cube. Giventhat AG = 1, determine the area of triangle AOP .
√ √
2 1 2
√ √
Answer: From AG = 1, we get that AE = and AC = . We note that triangle AOP is
3 324
located in the plane of rectangle ACGE . Since OP ‖ CG and O is halway between AC and EG , we
√ √
1 1 1 2 2
√ √
get that [ AOP ] = [ ACGE ]. Hence, [ AOP ] = ( )( ) = .
8 8 24
3 3
1A
√
E3
√FB
√2
O3
CHDP G