Difference between revisions of "2016 AMC 8 Problems/Problem 23"
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==Problem== | ==Problem== | ||
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Two congruent circles centered at points <math>A</math> and <math>B</math> each pass through the other circle's center. The line containing both <math>A</math> and <math>B</math> is extended to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree measure of <math>\angle CED</math>? | Two congruent circles centered at points <math>A</math> and <math>B</math> each pass through the other circle's center. The line containing both <math>A</math> and <math>B</math> is extended to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree measure of <math>\angle CED</math>? | ||
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Now, <math>\angle{CED}=m\angle{AEC}+m\angle{AEB}+m\angle{BED} = 30^{\circ}+60^{\circ}+30^{\circ} = 120^{\circ}</math>. Therefore, the answer is <math>\boxed{\textbf{(C) }\ 120}</math>. | Now, <math>\angle{CED}=m\angle{AEC}+m\angle{AEB}+m\angle{BED} = 30^{\circ}+60^{\circ}+30^{\circ} = 120^{\circ}</math>. Therefore, the answer is <math>\boxed{\textbf{(C) }\ 120}</math>. | ||
− | ==Video Solution== | + | == Video Solution 1 == |
+ | https://youtu.be/UZqVG5Q1liA?si=LDc8tMTnj1FMMlZc | ||
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+ | == Video Solution 2 == | ||
https://youtu.be/iGG_Hz-V6lU | https://youtu.be/iGG_Hz-V6lU | ||
− | + | == Video Solution 3== | |
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− | == Video Solution | ||
https://youtu.be/FDgcLW4frg8?t=968 | https://youtu.be/FDgcLW4frg8?t=968 | ||
− | + | == Video Solution 4 == | |
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− | ==Video Solution== | ||
https://youtu.be/nLlnMO6D5ek | https://youtu.be/nLlnMO6D5ek | ||
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==See Also== | ==See Also== | ||
{{AMC8 box|year=2016|num-b=22|num-a=24}} | {{AMC8 box|year=2016|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 16:29, 13 January 2025
Contents
[hide]Problem
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 ?
Solutions
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 .
Video Solution 1
https://youtu.be/UZqVG5Q1liA?si=LDc8tMTnj1FMMlZc
Video Solution 2
Video Solution 3
https://youtu.be/FDgcLW4frg8?t=968
Video Solution 4
See Also
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.