2005 Alabama ARML TST Problems/Problem 3
Problem
The difference between the areas of the circumcircle and incircle of an equilateral triangle is square units. Find the number of units in the length of a side of the triangle.
Solution
Let be the radius of the circumcircle and let be the radius of the incircle. Then we have . If the center of these two circles is , the vertices are and , and is the midpoint of side , triangle is a right triangle, and its hypotenuse has length and its shorter leg has length . Thus . (There are many other arguments to get to this conclusion; for instance, is also the centroid of the triangle and is a median, so trisects and .)
Then so and the side length of the triangle is equal to .
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 |