2008 AMC 10B Problems/Problem 10

Problem

Points A and B are on a circle of radius 5 and AB=6. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC?

(A) $\sqrt{10}$ (B) $\frac{7}{2}$ (C) $\sqrt{14}$

(D) $\sqrt{15}$ (E) $4$

Let the center of the circle be O. Draw lines

OA, OB, and OC.

OA=OB=5, since they are both radii.

OC bisects AB, and AB=6, so letting point D be

the intersection of AB and OC, AD=BD=3.

Also, OD=4. This means that CD=1.

Using the pythagorean theorem,

$AC=\sqrt{3^2+1^2}=\sqrt{10}$ .

Answer A is the correct answer.

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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