Difference between revisions of "2014 AIME I Problems/Problem 13"
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<cmath>\frac{15^2 \cdot 34^2}{d^2} + d^2 = 34^2</cmath> | <cmath>\frac{15^2 \cdot 34^2}{d^2} + d^2 = 34^2</cmath> | ||
− | <cmath>d^4 - 34^2 d^2 + 15^2 \cdot 34^2</cmath> | + | <cmath>d^4 - 34^2 d^2 + 15^2 \cdot 34^2 = 0</cmath> |
− | <cmath>(d^2 - 34 \cdot 25)(d^2 - 34 \cdot 9) 0</cmath> | + | <cmath>(d^2 - 34 \cdot 25)(d^2 - 34 \cdot 9) = 0</cmath> |
− | <math>34 \cdot 9</math> is extraneous, so <math>d^2 = 34 \cdot 25</math>. Since the area is d^2, we have it is equal to <math>34 \cdot 25 = \boxed{850}</math> | + | <math>d^2 = 34 \cdot 9</math> is extraneous, so <math>d^2 = 34 \cdot 25</math>. Since the area is d^2, we have it is equal to <math>34 \cdot 25 = \boxed{850}</math> |
+ | |||
+ | -Alexlikemath | ||
== See also == | == See also == | ||
{{AIME box|year=2014|n=I|num-b=12|num-a=14}} | {{AIME box|year=2014|n=I|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 18:58, 26 December 2020
Problem 13
On square , points , and lie on sides and respectively, so that and . Segments and intersect at a point , and the areas of the quadrilaterals and are in the ratio Find the area of square .
Solution
Notice that . This means passes through the center of the square.
Draw with on , on such that and intersects at the center of the square which I'll label as .
Let the area of the square be . Then the area of and the area of . This is because is perpendicular to (given in the problem), so is also perpendicular to . These two orthogonal lines also pass through the center of the square, so they split it into 4 congruent quadrilaterals.
Let the side length of the square be .
Draw and intersects at . .
The area of , so the area of .
Let . Then
Consider the area of .
Thus, .
Solving , we get .
Therefore, the area of
Lazy Solution
, a multiple of . In addition, , which is . Therefore, we suspect the square of the "hypotenuse" of a right triangle, corresponding to and must be a multiple of . All of these triples are primitive:
The sides of the square can only equal the longer leg, or else the lines would have to extend outside of the square. Substituting :
Thus, is the only valid answer.
Solution 3
Continue in the same way as solution 1 to get that POK has area 3a, and OK = d/10. You can then find PK has length 3/2.
Then, if we drop a perpendicular from H to BC get L, We get HLF ~ OPK.
Thus, , and we know HL = d, and HF = 34. Thus, we can set up an equation in terms of d using the Pythagorean theorem.
is extraneous, so . Since the area is d^2, we have it is equal to
-Alexlikemath
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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.