Difference between revisions of "2011 AIME I Problems/Problem 2"
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Revision as of 21:42, 18 March 2011
Problem
In rectangle , and . Points and lie inside rectangle so that ,,,, and line intersects segment . The length can be expressed in the form , where ,, and are positive integers and is not divisible by the square of any prime. Find .
Solution
Let us call the point where intersects point , and the point where intersects point . Since angles and are both right angles, and angles and are congruent due to parallelism, right triangles and are similar. This implies that . Since , . ( is the same as because they are opposite sides of a rectangle.) Now, we have a system:
Solving this system (easiest by substitution), we get that:
Using the Pythagorean Theorem, we can solve for the remaining sides of the two right triangles:
and
Notice that adding these two sides would give us twelve plus the overlap . This means that:
Since isn't divisible by any perfect square, our answer is: