Difference between revisions of "2016 AMC 8 Problems/Problem 23"
(→Solution 1) |
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label("</math>D<math>", D, SE); | label("</math>D<math>", D, SE); | ||
label("</math>E<math>", E, N); | label("</math>E<math>", E, N); | ||
− | </asy>% | + | </asy>%</math> |
− | we see that < | + | 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 |
==Solution 2== | ==Solution 2== |
Revision as of 21:19, 28 May 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:
$<asy
label("$ (Error compiling LaTeX. ! Missing $ inserted.)DE$", E, N); </asy>%$ (Error compiling LaTeX. ! Missing $ inserted.)
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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