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Difference between revisions of "2005 Alabama ARML TST Problems/Problem 3"
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[[Category:Intermediate Geometry Problems]]

Revision as of 11:43, 11 December 2007

Problem

The difference between the areas of the circumcircle and incircle of an equilateral triangle is $300\pi$ square units. Find the number of units in the length of a side of the triangle.

Solution

Let $R$ be the radius of the circumcircle and let $r$ be the radius of the incircle. Then we have $R^2-r^2=300$. If the center of these two circles is $O$, the vertices are $A, B$ and $C$, and $M$ is the midpoint of side $AB$, triangle $\triangle AMO$ is a $30^\circ-60^\circ-90^\circ$ right triangle, and its hypotenuse has length $R$ and its shorter leg has length $r$. Thus $R = 2r$. (There are many other arguments to get to this conclusion; for instance, $O$ is also the centroid of the triangle and $COM$ is a median, so $O$ trisects $CO$ and $R = CO = 2OM = 2r$.)

Then $4r^2 - r^2 = 300$ so $r = 10$ and the side length of the triangle is equal to $10\sqrt 3$.

See Also

2005 Alabama ARML TST (Problems)
Preceded by:
Problem 2 		Followed by:
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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