Difference between revisions of "2016 AMC 8 Problems/Problem 23"
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label("<math>D</math>", D, SE); | label("<math>D</math>", D, SE); | ||
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we see that <math>\triangle EAB</math> is equilateral as each side is the radius of one of the two circles. Therefore, <math>\overarc{EB}=m\angle EAB=60^\circ</math>. Therefore, since it is an inscribed angle | we see that <math>\triangle EAB</math> is equilateral as each side is the radius of one of the two circles. Therefore, <math>\overarc{EB}=m\angle EAB=60^\circ</math>. Therefore, since it is an inscribed angle |
Revision as of 17:44, 4 July 2019
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:
D</math>", D, SE); label("<math>E</math>", E, N); (Error compiling LaTeX. 23e67869e1252b9ed56563a45ccdf2c266b7c77a.asy: 5.3: syntax error error: could not load module '23e67869e1252b9ed56563a45ccdf2c266b7c77a.asy')
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 |
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