Difference between revisions of "1996 AHSME Problems/Problem 18"
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+ | ==Problem== | ||
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+ | A circle of radius <math>2</math> has center at <math>(2,0)</math>. A circle of radius <math>1</math> has center at <math>(5,0)</math>. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the <math>y</math>-intercept of the line? | ||
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+ | <math> \text{(A)}\ \sqrt{2}/4\qquad\text{(B)}\ 8/3\qquad\text{(C)}\ 1+\sqrt 3\qquad\text{(D)}\ 2\sqrt 2\qquad\text{(E)}\ 3 </math> | ||
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+ | ==Solution== | ||
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+ | The two circles are tangent to eaech other at the point <math>(4,0)</math>, since it is both <math>2</math> units from <math>(2,0)</math> and <math>1</math> unit from <math>(5,0)</math>. | ||
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+ | Label the x-intercept of the common tangent line <math>A</math>, and label the y-intercept of the common tangent <math>B</math>. Triangle <math>\triangle OAB</math> is a right triangle at the origin. | ||
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+ | Label <math>D</math> the point of tangency to the first, big circle, and label <math>E</math> the point of tangency to the small circle. <math>\angle D</math> and <math>\angle E</math> are both right angles as well. | ||
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+ | Label the center of the big circle <math>F</math>, and the small circle <math>G</math>. | ||
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+ | <math>\triangle DAF \sim \triangle EAG \sim \triangle OAB</math> because they each have one right angle, and also ahve a common angle <math>A</math>. | ||
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+ | The y-intercept is the length <math>OB</math>. | ||
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+ | We know that <math>DF = 2</math> and <math>EG = 1</math> because they are radii of circles. From similarity, <math>\frac{DF}{EG} = \frac{FA}{GA}</math>. Thus, <math>FA = 2GA</math>. | ||
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+ | Looking at line <math>FGA</math>, we know that <math>FG + GA = FA</math>. | ||
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+ | Thus, <math>FG + GA = 2GA</math>, meaning <math>FG = GA</math>. Since <math>FG = 3</math> because it's the distance of the centers of the circles, so <math>GA = 3</math> as well. This gives point <math>A</math> as <math>(8,0)</math>. | ||
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+ | Since <math>\triangle EAG</math> is a right triangle with <math>GA = 3</math> and <math>GE = 1</math>, we know that <math>AE = \sqrt{GA^2 - GE^2} = \sqrt{8}</math> | ||
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+ | Thus, all triangles are <math>1:\sqrt{8}:3</math> triangles. | ||
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+ | <math>\triangle OAC</math> has as its middle side <math>OA</math>, which is <math>8</math>. Thus, all sides are scaled up by a factor of <math>\sqrt{8}</math>, and <math>OB</math>, the shortest side, is <math>\sqrt{8}</math>. This means the y-intercept is <math>\sqrt{8} = 2\sqrt{2}</math>, and the answer is <math>\boxed{D}</math>. | ||
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==See also== | ==See also== | ||
{{AHSME box|year=1996|num-b=17|num-a=19}} | {{AHSME box|year=1996|num-b=17|num-a=19}} |
Revision as of 18:50, 19 August 2011
Problem
A circle of radius has center at . A circle of radius has center at . A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the -intercept of the line?
Solution
The two circles are tangent to eaech other at the point , since it is both units from and unit from .
Label the x-intercept of the common tangent line , and label the y-intercept of the common tangent . Triangle is a right triangle at the origin.
Label the point of tangency to the first, big circle, and label the point of tangency to the small circle. and are both right angles as well.
Label the center of the big circle , and the small circle .
because they each have one right angle, and also ahve a common angle .
The y-intercept is the length .
We know that and because they are radii of circles. From similarity, . Thus, .
Looking at line , we know that .
Thus, , meaning . Since because it's the distance of the centers of the circles, so as well. This gives point as .
Since is a right triangle with and , we know that
Thus, all triangles are triangles.
has as its middle side , which is . Thus, all sides are scaled up by a factor of , and , the shortest side, is . This means the y-intercept is , and the answer is .
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |