2006 AMC 10A Problems/Problem 16
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Problem
A circle of radius 1 is tangent to a circle of radius 2. The sides of are tangent to the circles as shown, and the sides
and
are congruent. What is the area of
?
Solution
Note that . Using the first pair of similar triangles, we write the proportion:
![$\frac{AO_1}{AO_2} = \frac{DO_1}{EO_2} \Longrightarrow \frac{AO_1}{AO_1 + 3} = \frac{1}{2} \Longrightarrow AO_1 = 3$](http://latex.artofproblemsolving.com/1/0/a/10ada06cd47acf9909068e7654b0ddc0944a8f67.png)
By the Pythagorean Theorem we have that .
Now using ,
![$\frac{AD}{AF} = \frac{DO_1}{CF} \Longrightarrow \frac{2\sqrt{2}}{8} = \frac{1}{CF} \Longrightarrow CF = 2\sqrt{2}$](http://latex.artofproblemsolving.com/7/c/5/7c56d91b20873ea411c0000555ca04283721060d.png)
The area of the triangle is .
See also
2006 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 | ||
All AMC 10 Problems and Solutions |
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