Difference between revisions of "1995 AHSME Problems/Problem 26"
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== Solution == | == Solution == | ||
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+ | '''Solution 1''' | ||
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Let the radius of the circle be <math>r</math> and let <math>x=\overline{OE}</math>. | Let the radius of the circle be <math>r</math> and let <math>x=\overline{OE}</math>. | ||
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Thus, the area of the circle is <math>\pi r^2 = 24\pi \Rightarrow C</math>. | Thus, the area of the circle is <math>\pi r^2 = 24\pi \Rightarrow C</math>. | ||
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+ | '''Solution 2''' | ||
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+ | Let the radius of the circle be <math>r</math>. | ||
+ | |||
+ | We can see that <math>\triangle CFD</math> has a right angle at <math>F</math> and that <math>\triangle EOD</math> has a right angle at <math>O</math>. | ||
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+ | Both triangles also share <math>\angle ODE</math>, so <math>\triangle CFD</math> and <math>\triangle EOD</math> are [[Similar(geometry)|similar]]. | ||
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+ | This means that <math>\frac {DE}{OD}=\frac {CD}{FD}</math>. | ||
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+ | So, <math>\frac {6}{r}=\frac {2r}{8}</math>. Simplifying, <math>r^2 = 24</math>. | ||
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+ | This means the area is <math>24\pi \Rightarrow C</math>. | ||
== See also == | == See also == |
Revision as of 20:19, 6 November 2013
Problem
In the figure, and are diameters of the circle with center , , and chord intersects at . If and , then the area of the circle is
Solution
Solution 1
Let the radius of the circle be and let .
By the Pythagorean Theorem, .
By Power of a point, .
Adding these equations yields .
Thus, the area of the circle is .
Solution 2
Let the radius of the circle be .
We can see that has a right angle at and that has a right angle at .
Both triangles also share , so and are similar.
This means that .
So, . Simplifying, .
This means the area is .
See also
1995 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.