2016 AMC 8 Problems/Problem 23
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]
pair A, B, C, D, E;
A = (0,0);
B = (10,0);
C = (-10,0);
D = (20,0);
E = (5, 8.75);
draw(Circle(A, 10));
draw(Circle(B, 10));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
draw(C--D);
draw(A--E);
draw(B--E);
draw(C--E);
draw(D--E);
label("", A, SW);
label("", B, SE);
label("", C, SW);
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, 
. So, in 
, 
, and 
. Our answer is 
.
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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