Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 1"
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==Solution== | ==Solution== | ||
− | {{ | + | Let the radius of the third circle be <math>r</math>. The side lengths of the triangle are <math>10</math>, <math>3+r</math>, and <math>7+r</math>. From Heron's Formula, <math>84=\sqrt{(10+r)(r)(7)(3)}</math>, or <math>84*84=r(10+r)*21</math>, or <math>84*4=r(10+r)</math>. <math>84*4=14*24</math>, so <math>r=14</math>. Thus the area of the circle is <math>\boxed{196}\pi</math>. |
− | ==See | + | ==See Also== |
+ | {{Mock AIME box|year=Pre 2005|n=3|before=First Question|num-a=2}} |
Latest revision as of 09:22, 4 April 2012
Problem
Three circles are mutually externally tangent. Two of the circles have radii and . If the area of the triangle formed by connecting their centers is , then the area of the third circle is for some integer . Determine .
Solution
Let the radius of the third circle be . The side lengths of the triangle are , , and . From Heron's Formula, , or , or . , so . Thus the area of the circle is .
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |