Difference between revisions of "2015 AIME I Problems/Problem 7"
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This gives that <math>AM = 2 \cdot AN = 2 \cdot \frac{3\sqrt{11}}{\sqrt5}</math> and <math>ME = \sqrt5 \cdot MP = \sqrt5 \cdot (ML - EG) = \sqrt5 \cdot (ML - (EC - (GF + FC))</math> | This gives that <math>AM = 2 \cdot AN = 2 \cdot \frac{3\sqrt{11}}{\sqrt5}</math> and <math>ME = \sqrt5 \cdot MP = \sqrt5 \cdot (ML - EG) = \sqrt5 \cdot (ML - (EC - (GF + FC))</math> | ||
+ | |||
+ | == See also == | ||
+ | {{AIME box|year=2015|n=I|num-b=6|num-a=8}} | ||
+ | {{MAA Notice}} | ||
+ | [[Category:Introductory Geometry Problems]] |
Revision as of 17:59, 20 March 2015
Problem
7. In the diagram below, is a square. Point is the midpoint of . Points and lie on , and and lie on and , respectively, so that is a square. Points and lie on , and and lie on and , respectively, so that is a square. The area of is 99. Find the area of .
Solution
We begin by denoting the length , giving us and . Since angles and are complimentary, we have that (and similarly the rest of the triangles are triangles). We let the sidelength of be , giving us:
and .
Since ,
,
Solving for in terms of yields .
We now use the given that , implying that . We also draw the perpendicular from E to ML and label the point of intersection P.
This gives that and
See also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.