Difference between revisions of "2008 AMC 10B Problems/Problem 10"

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==Problem==
 
==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?
 
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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(A) <math>\sqrt{10}</math> (B) <math>\frac{7}{2}</math> (C) <math>\sqrt{14}</math>
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(D) <math>\sqrt{15}</math> (E) <math>4</math>
  
 
==Solution==
 
==Solution==

Revision as of 17:42, 10 August 2008

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) $4$

Solution

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.

See also

2008 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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 AMC 10 Problems and Solutions
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