Difference between revisions of "2007 AIME I Problems/Problem 9"
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== Problem == | == Problem == | ||
| − | In [[right triangle]] <math>ABC</math> with [[right angle]] <math>C</math>, <math>CA = 30</math> and <math>CB = 16</math>. Its legs <math>CA</math> and <math>CB</math> are extended beyond <math>A</math> and <math>B</math>. [[Point]]s <math>O_1</math> and <math>O_2</math> lie in the exterior of the triangle and are the centers of two [[circle]]s with equal [[radii]]. The circle with center <math>O_1</math> is tangent to the [[hypotenuse]] and to the extension of leg <math>CA</math>, the circle with center <math>O_2</math> is [[tangent]] to the hypotenuse and to the extension of [[leg]] <math>CB</math>, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as <math>p/q</math>, where <math>p</math> and <math>q</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>p+q</math>. | + | In [[right triangle]] <math>ABC</math> with [[right angle]] <math>C</math>, <math>CA = 30</math> and <math>CB = 16</math>. Its legs <math>CA</math> and <math>CB</math> are extended beyond <math>A</math> and <math>B</math>. [[Point]]s <math>O_1</math> and <math>O_2</math> lie in the exterior of the triangle and are the centers of two [[circle]]s with equal [[radius|radii]]. The circle with center <math>O_1</math> is tangent to the [[hypotenuse]] and to the extension of leg <math>CA</math>, the circle with center <math>O_2</math> is [[tangent]] to the hypotenuse and to the extension of [[leg]] <math>CB</math>, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as <math>p/q</math>, where <math>p</math> and <math>q</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>p+q</math>. |
== Solution == | == Solution == | ||
| − | Let the point where CB's extension hits the circle be D, and the point where the hypotenuse hits that circle be E. Clearly EB=DB. Let <math>EB=x</math>. Draw the two [[perpendicular]] radii to D and E. Now we have a [[cyclic quadrilateral]]. Let the radius be length <math>r</math>. We see that since the cosine of angle ABC is <math>\frac{15}{17}</math> the cosine of angle EBD is <math>-\frac{15}{17}</math>. Since the measure of the angle opposite to EBD is the [[complement]] of this one, its cosine is <math>\frac{15}{17}</math>. Using the law of cosines, we see that <math>x^{2}+x^{2}+\frac{30x^{2}}{17}=r^{2}+r^{2}-\frac{30r^{2}}{17}</math> | + | {{image}} |
| + | Let the point where CB's extension hits the circle be D, and the point where the hypotenuse hits that circle be E. Clearly <math>EB=DB</math>. Let <math>EB=x</math>. Draw the two [[perpendicular]] radii to D and E. Now we have a [[cyclic quadrilateral]]. Let the radius be length <math>r</math>. We see that since the cosine of angle ABC is <math>\frac{15}{17}</math> the cosine of angle EBD is <math>-\frac{15}{17}</math>. Since the measure of the angle opposite to EBD is the [[complement]] of this one, its cosine is <math>\frac{15}{17}</math>. Using the law of cosines, we see that <math>x^{2}+x^{2}+\frac{30x^{2}}{17}=r^{2}+r^{2}-\frac{30r^{2}}{17}</math> | ||
This tells us that <math>r=4x</math>. | This tells us that <math>r=4x</math>. | ||
Revision as of 17:58, 15 March 2007
Problem
In right triangle 
with right angle 
, 
and 
. Its legs 
and 
are extended beyond 
and 
. Points 
and 
lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center is tangent to the hypotenuse and to the extension of leg , the circle with center is tangent to the hypotenuse and to the extension of leg , and the circles are externally tangent to each other. The length of the radius either circle can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Let the point where CB's extension hits the circle be D, and the point where the hypotenuse hits that circle be E. Clearly 
. Let 
. Draw the two perpendicular radii to D and E. Now we have a cyclic quadrilateral. Let the radius be length 
. We see that since the cosine of angle ABC is 
the cosine of angle EBD is 
. Since the measure of the angle opposite to EBD is the complement of this one, its cosine is . Using the law of cosines, we see that 
This tells us that 
.
Now look at the other end of the hypotenuse. Call the point where CA hits the circle F and the point where the hypotenuse hits the circle G. Draw the radii to F and G and we have cyclic quadrilaterals once more.
Using the law of cosines again, we find that the length of our tangents is 
. Note that if we connect the centers of the circles we have a rectangle with sidelengths 8x and 4x. So, 8x+2.4x+x=34. Solving we find that 
so our answer is 737.
See also
| 2007 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
