2016 AMC 8 Problems/Problem 23
Two congruent circles centered at points and each pass through the other circle's center. The line containing both and is extended to intersect the circles at points and . The circles intersect at two points, one of which is . What is the degree measure of ?
Solution 1
Drawing the diagram:
label("$D$", D, SE); label("$E$", E, N); (Error compiling LaTeX. de252c60d1f2fe7247019fff9f040e897514e15b.asy: 5.6: no matching function 'label(string, <overloaded>, pair)' de252c60d1f2fe7247019fff9f040e897514e15b.asy: 5.14: use of variable 'D' is ambiguous)
we see that is equilateral as each side is the radius of one of the two circles. Therefore, . Therefore, since it is an inscribed angle
Solution 2
As in Solution 1, observe that is equilateral. Therefore, . Since is a straight line, we conclude that . Since (both are radii of the same circle), is isosceles, meaning that . Similarly, .
Now, . Therefore, the answer is .
2016 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.