1999 AIME Problems/Problem 4
From AoPSWiki
Problem
The two squares shown share the same center 
and have sides of length 1. The length of 
is 
and the area of octagon 
is 
where 
and 
are relatively prime positive integers. Find 
Contents |
Solution
Solution 1
Define the two possible distances from one of the labeled points and the corners of the square upon which the point lies as 
and 
. The area of the octagon (by subtraction of areas) is 
.
By the Pythagorean theorem,
Also,
Substituting,
Thus, the area of the octagon is 
, so 
.
Solution 2
Each of the triangle 
, 
, 
, etc. are congruent, and their areas are 
, since the area of a triangle is 
, so the area of all 
of them is 
and the answer is 
.
See also
| 1999 AIME (Problems • Resources) | ||
| Preceded by Problem 3 | Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
