2016 AMC 8 Problems/Problem 23: Difference between revisions
| Line 7: | Line 7: | ||
[asy] | [asy] | ||
label("<math>D</math>", D, SE); | label("<math>D</math>", D, SE); | ||
label("<math>E</math>", E, N); | label("<math>E</math>", E, N); | ||
Revision as of 09:52, 3 November 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("", D, SE);
label("", E, N);
[\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 | |
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