2008 AMC 10B Problem 10

Problem 10 of 25EasierGeometry

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?

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Solution

Let

O

be the center and

D

the midpoint of

AB.

Then

OD\perp AB

with

AD=3,

so

OD=\sqrt{5^2-3^2}=4.

Since

C

is the midpoint of the minor arc,

O,

D,

C

are collinear and

DC=OC-OD=5-4=1.

Then

AC=\sqrt{AD^2+DC^2}

=\sqrt{3^2+1^2}

=\sqrt{10}.

Thus, the correct answer is A.

More practice

Concepts: chord · Pythagorean Theorem · circle

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.