HMMT 二月 2006 · GEN1 赛 · 第 3 题
HMMT February 2006 — GEN1 Round — Problem 3
题目详情
英文原题
- A square and an equaliteral triangle together have the property that the area of eachis the perimeter of the other. Find the square’s area.
解析
英文解析
- A square and an equaliteral triangle together have the property that the area of eachis the perimeter of the other. Find the square’s area.
√
Answer: 12 43
Solution: Let s, t be the side lengths of the square and triangle, respectively. Then
√
3 t 2
4 s = , and so
( )4
√ 2
2 4
3 t 3 t
3 t = s = = ,2
16 28
√
√
3 8 233
so that t = 2 and s = 3 t = 3 2 = 12 4 .8