HMMT 十一月 2019 · THM 赛 · 第 9 题

HMMT November 2019 — THM Round — Problem 9

For dinner, Priya is eating grilled pineapple spears. Each spear is in the shape of the quadrilateral P IN E , ◦ with P I = 6cm, IN = 15cm, N E = 6cm, EP = 25cm, and ∠ N EP + ∠ EP I = 60 . What is the area of 2 each spear, in cm ?

解析

For dinner, Priya is eating grilled pineapple spears. Each spear is in the shape of the quadrilateral P IN E , ◦ with P I = 6cm, IN = 15cm, N E = 6cm, EP = 25cm, and ∠ N EP + ∠ EP I = 60 . What is the area of 2 each spear, in cm ? 3 Proposed by: Milan Haiman √ 100 3 Answer: 3 We consider a configuration composed of 2 more quadrilaterals congruent to P IN E . Let them be ′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′ ′′ ′ ′′ ′ ′′ P I N E , with E = P and N = I , and P I N E with P = E, E = P , N = I , and I = N . ′ ′′ ′′ ′ ′ ′′ Notice that this forms an equilateral triangle of side length 25 since ∠ P P P = ∠ P P P = ∠ P P P = ◦ ′ ′′