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Difference between revisions of "2016 AMC 8 Problems/Problem 23"

m (→‎Solution 1)
(→‎Solution 1)
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Drawing the diagram:
 
Drawing the diagram:
  
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<asy
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<math><asy
  
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label("<math>D</math>", D, SE);
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label("</math>D<math>", D, SE);
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label("<math>E</math>", E, N);
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label("</math>E<math>", E, N);
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</asy>
+
</asy>%
  
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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
+
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$. Therefore, since it is an inscribed angle
  
 
==Solution 2==
 
==Solution 2==

Revision as of 20:18, 28 May 2019

Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$?

$\textbf{(A) }90\qquad\textbf{(B) }105\qquad\textbf{(C) }120\qquad\textbf{(D) }135\qquad \textbf{(E) }150$

Solution 1

Drawing the diagram:

$<asy

label("$ (Error compiling LaTeX. Unknown error_msg)D$", D, SE); label("$E$", E, N); </asy>%

we see that$ (Error compiling LaTeX. Unknown error_msg)\triangle EAB$is equilateral as each side is the radius of one of the two circles. Therefore,$\overarc{EB}=m\angle EAB=60^\circ$. Therefore, since it is an inscribed angle

Solution 2

As in Solution 1, observe that $\triangle{EAB}$ is equilateral. Therefore, $m\angle{AEB}=m\angle{EAB}=m\angle{EBA} = 60^{\circ}$. Since $CD$ is a straight line, we conclude that $m\angle{EBD} = 180^{\circ}-60^{\circ}=120^{\circ}$. Since $BE=BD$ (both are radii of the same circle), $\triangle{BED}$ is isosceles, meaning that $m\angle{BED}=m\angle{BDE}=30^{\circ}$. Similarly, $m\angle{AEC}=m\angle{ACE}=30^{\circ}$.

Now, $\angle{CED}=m\angle{AEC}+m\angle{AEB}+m\angle{BED} = 30^{\circ}+60^{\circ}+30^{\circ} = 120^{\circ}$. Therefore, the answer is $\boxed{\textbf{(C) }\ 120}$.


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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