2006 AIME I Problems/Problem 6
Problem
Square has sides of length 1. Points and are on and respectively, so that is equilateral. A square with vertex has sides that are parallel to those of and a vertex on The length of a side of this smaller square is where and are positive integers and is not divisible by the square of any prime. Find
Solution
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Call the vertices of the new square A', B', C', and D', in relation to the vertices of , and define to be one of the sides of that square. Since the sides are parallel, by corresponding angles and AA~ we know that triangles and are similar. Thus, the sides are proportional: . Simplifying, we get that .
is degrees, so . Thus, , so . Since is equilateral, . is a , so . Substituting back into the equation from the beginning, we get , so . Therefore, , and .
See also
2006 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |