Difference between revisions of "2012 AIME I Problems/Problem 13"
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− | Reinterpret the problem in the following manner. Equilateral triangle <math>ABC</math> has a point <math>X</math> on the interior such that <math>AX = 5,</math> <math>BX = 4,</math> and <math>CX = 3.</math> A <math>60^o</math> clockwise rotation about vertex <math>A</math> maps <math>X</math> to <math>X'</math> and <math>C</math> to <math>C'.</math> Note that angle <math>XAX'</math> is <math>60</math> and <math>XA = X'A = 5</math> which tells us that triangle <math>XAX'</math> is equilateral and that <math>XX' = 5.</math> We now notice that <math>XC = 3</math> and <math>X'C = 4</math> which tells us that angle <math>XCX'</math> is <math>90</math> because there is a <math>3</math>-<math>4</math>-<math>5</math> Pythagorean triple. Now note that <math>\angle ABC + \angle ACB = 120</math> and <math>\angle XCA + \angle XBA = 90,</math> so <math>\angle XCB+\angle XBC = 30</math> and <math>\angle BXC = 150.</math> Applying the law of cosines on triangle <math>BXC</math> yields | + | Reinterpret the problem in the following manner. Equilateral triangle <math>ABC</math> has a point <math>X</math> on the interior such that <math>AX = 5,</math> <math>BX = 4,</math> and <math>CX = 3.</math> A <math>60^o</math> counter-clockwise rotation about vertex <math>A</math> maps <math>X</math> to <math>X'</math> and <math>C</math> to <math>C'.</math> Note that angle <math>XAX'</math> is <math>60</math> and <math>XA = X'A = 5</math> which tells us that triangle <math>XAX'</math> is equilateral and that <math>XX' = 5.</math> We now notice that <math>XC = 3</math> and <math>X'C = 4</math> which tells us that angle <math>XCX'</math> is <math>90</math> because there is a <math>3</math>-<math>4</math>-<math>5</math> Pythagorean triple. Now note that <math>\angle ABC + \angle ACB = 120</math> and <math>\angle XCA + \angle XBA = 90,</math> so <math>\angle XCB+\angle XBC = 30</math> and <math>\angle BXC = 150.</math> Applying the law of cosines on triangle <math>BXC</math> yields |
<cmath>BC^2 = BX^2+CX^2 - 2 \cdot BX \cdot CX \cdot \cos(150) = 4^2+3^2-24 \cdot \frac{-\sqrt{3}}{2} = 25+12\sqrt{3}</cmath> | <cmath>BC^2 = BX^2+CX^2 - 2 \cdot BX \cdot CX \cdot \cos(150) = 4^2+3^2-24 \cdot \frac{-\sqrt{3}}{2} = 25+12\sqrt{3}</cmath> |
Revision as of 21:01, 27 March 2012
Problem 13
Three concentric circles have radii and An equilateral triangle with one vertex on each circle has side length The largest possible area of the triangle can be written as where and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find
Solution
Reinterpret the problem in the following manner. Equilateral triangle has a point on the interior such that and A counter-clockwise rotation about vertex maps to and to Note that angle is and which tells us that triangle is equilateral and that We now notice that and which tells us that angle is because there is a -- Pythagorean triple. Now note that and so and Applying the law of cosines on triangle yields
and thus the area of equals
so our final answer is
See also
2012 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 |