HMMT 二月 2003 · 几何 · 第 7 题
HMMT February 2003 — Geometry — Problem 7
题目详情
英文原题
- Let RST U V be a regular pentagon. Construct an equilateral triangle P RS with point
P inside the pentagon. Find the measure (in degrees) of angle P T V .
解析
英文解析
- Let RST U V be a regular pentagon. Construct an equilateral triangle P RS with point
P inside the pentagon. Find the measure (in degrees) of angle P T V .
Solution: 6
° ° °
6 6 6
We have P RV = SRV − SRP = 108 − 60 = 48 . Since P R = RS = RV , triangle
° °
6 6 6
P RV is isosceles, so that V P R = RV P = (180 − P RV ) / 2 = 66 . Likewise, we
°
have T P S = 66 , so that 6
° ° ° ° ° °
6 6 6 6
T P V = 360 − ( V P R + RP S + SP T ) = 360 − (66 + 60 + 66 ) = 168 .
6 6
Finally, by symmetry, triangle P T V is isosceles ( P T = T V ), so P T V = T V P =
° °
(180 − T P V ) / 2 = 6 . (See the figure.)6
V TU
R SP