Difference between revisions of "2015 AIME I Problems/Problem 7"
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− | + | In the diagram below, <math>ABCD</math> is a square. Point <math>E</math> is the midpoint of <math>\overline{AD}</math>. Points <math>F</math> and <math>G</math> lie on <math>\overline{CE}</math>, and <math>H</math> and <math>J</math> lie on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, so that <math>FGHJ</math> is a square. Points <math>K</math> and <math>L</math> lie on <math>\overline{GH}</math>, and <math>M</math> and <math>N</math> lie on <math>\overline{AD}</math> and <math>\overline{AB}</math>, respectively, so that <math>KLMN</math> is a square. The area of <math>KLMN</math> is 99. Find the area of <math>FGHJ</math>. | |
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Revision as of 22:56, 9 May 2020
Problem
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 1
Let us find the proportion of the side length of and . Let the side length of and the side length of .
Now, examine . We know , and triangles and are similar to since they are triangles. Thus, we can rewrite in terms of the side length of .
Now examine . We can express this length in terms of since . By using similar triangles as in the first part, we have
Now, it is trivial to see that
Solution 2
We begin by denoting the length , giving us and . Since angles and are complementary, 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 to and label the point of intersection :
This gives that and
Since = , we get
So our final answer is
Solution 3
This is a relatively quick solution but a fakesolve. We see that with a ruler, cm and cm. Thus if corresponds with an area of , then ('s area) would correspond with - aops5234
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 |
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