1992 AHSME Problems/Problem 11
Problem
The ratio of the radii of two concentric circles is . If is a diameter of the larger circle, is a chord of the larger circle that is tangent to the smaller circle, and , then the radius of the larger circle is
Solution (Similarity)
We are given that is tangent to the smaller circle. Using that, we know where the circle intersects , it creates a right triangle. We can also point out that since is the diameter of the bigger circle and triangle is inscribed the semi-circle, that angle is a right angle. Therefore, we have similar triangles. Let's label the center of the smaller circle (which is also the center of the larger circle) as . Let's also label the point where the smaller circle intersects as . So is similar to . Since is the radius of the smaller circle, call the length and since is the radius of the bigger circle, call that length . The diameter, is . So,
See also
1992 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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.