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
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 .
Solution 2 -SweetMango77
We know that is equilateral, because all of its sides are congruent radii. Because point is the center of a circle, is at the border of a circle, and and are points on the edge of that circle, $$ (Error compiling LaTeX. ! Missing $ inserted.)m\angle{ACB}=\frac{1}{2}\cdot m\angle{EAB}=\frac{1}{2}\cdot60^{\circ}=30^{\circ}\triangle{CED}\angle{CED}=180^{\circ}-2\cdot30^{\circ}=\boxed{\text{(A)}\; 120}$ degrees.
2016 AMC 8 (Problems • Answer Key • Resources) | ||
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Followed by Problem 24 | |
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