HMMT 二月 2003 · 几何 · 第 7 题

HMMT February 2003 — Geometry — Problem 7

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

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 ◦ 6 have T P S = 66 , so that ◦ ◦ ◦ ◦ ◦ ◦ 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 = ◦ ◦ 6 (180 − T P V ) / 2 = 6 . (See the figure.) U V T P R S