2016 AMC 8 Problem 23

Question

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?

90

105

120

135

150

The centers and one intersection point form an equilateral triangle.

Use the fact that angles subtending diameters are right angles.

Click to load, then click again to play

We know that

AE = EB = AB

since they are all radii of congruent circles, so they form an equilateral triangle, which means that

\angle AEB = 60^{\circ}.

The segments

\overline{DB}

and

\overline{AC}

are diameters, so

\angle DEB=\angle AEC=90^\circ.

Therefore,

\begin{aligned} \angle CED &= \angle DEB+\angle AEC\\ &\qquad-\angle AEB\\ &=90^\circ+90^\circ-60^\circ\\ &=120^\circ. \end{aligned}

Thus, C is the correct answer.

Problem 23 in other years