HMMT 二月 2000 · 几何 · 第 6 题
HMMT February 2000 — Geometry — Problem 6
题目详情
- What is the area of the largest ir le on tained in an equilateral triangle of area 8 3?
y 22 x
英文原题
What is the area of the largest ir le on tained in an equilateral triangle of area 8 p 3? 7. Let AB C b e a triangle ins rib ed in the ellipse x 2 4 + y 2 9 = 1. If its en troid is the origin (0,0), nd its area. 8. A sphere is ins rib ed inside a p yramid with a square as a base whose heigh t is p 15 2 times the length of one edge of the base. A ub e is ins rib ed inside the sphere. What is the ratio of the v olume of the p yramid to the v olume of the ub e? 9. Ho w man y hexagons are in the gure b elo w with v erti es on the giv en v erti es? (Note that a hexagon need not b e on v ex, and edges ma y ross!)
解析
英文解析
- The area of an equilateral triangle with side length s is s . Therefore, the side lengthp p 4
is 4 2 and heigh t h = 2 6. No w, if r is the radius of the ins rib ed ir le, then r = ,hsin e we have an equilateral triangle. Th us the area is r = .283
2 y 3
0 0