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
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. 
