HMMT 十一月 2012 · THM 赛 · 第 6 题
HMMT November 2012 — THM Round — Problem 6
题目详情
英文原题
[3] A rectangular piece of paper with vertices ABCD is being cut by a pair of scissors. The pair of
scissors starts at vertex A, and then cuts along the angle bisector of DAB until it reaches another edge
of the paper. One of the two resulting pieces of paper has 4 times the area of the other piece. W hat
is the ratio of the longer side of the original paper to the shorter side?
解析
英文解析
- [ 3 ] A rectangular piece of paper with vertices ABCD is being cut by a pair of scissors. The pair ofscissors starts at vertex A , and then cuts along the angle bisector of DAB until it reaches another edgeof the paper. One of the two resulting pieces of paper has 4 times the area of the other piece. Whatis the ratio of the longer side of the original paper to the shorter side?
Answer: Without loss of generality, let AB > AD , and let x = AD , y = AB . Let the cut along 5
the angle bisector of ∠ DAB meet CD at E . Note that ADE is a 45-45-90 triangle, so DE = AD = x ,2
x x 2
and EC = y − x . Now, [ ADE ] = , and [ AECB ] = x ( y − ) = 4[ ADE ]. Equating and dividing both
2 2
x 5
sides by x , we find that 2 x = y − , so y/x = .
2 2