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?
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Points <math>A</math> and <math>B</math> are on a circle of radius <math>5</math> and <math>AB=6</math>. Point <math>C</math> is the [[midpoint]] of the minor arc <math>AB</math>. What is the length of the line segment <math>AC</math>?
  
(A) <math>\sqrt{10}</math> (B) <math>\frac{7}{2}</math> (C) <math>\sqrt{14}</math>
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<math>\mathrm{(A)}\ \sqrt{10}\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ \sqrt{14}\qquad\mathrm{(D)}\ \sqrt{15}\qquad\mathrm{(E)}\ 4</math>
 
 
(D) <math>\sqrt{15}</math> (E) <math>4</math>
 
  
 
==Solution==
 
==Solution==
Let the center of the circle be O. Draw lines
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Let the center of the circle be <math>O</math>, and let <math>D</math> be the intersection of <math>\overline{AB}</math> and <math>\overline{OC}</math> (then <math>D</math> is the midpoint of <math>\overline{AB}</math>). <math>OA=OB=5</math>, since they are both radii.
 
 
OA, OB, and OC.
 
 
 
OA=OB=5, since they are both radii.
 
  
OC bisects AB, and AB=6, so letting point D be
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By the [[Pythagorean Theorem]], <math>OD = \sqrt{OA^2 - DA^2} = 4</math>, and by subtraction, <math>CD=OC - OD = 1</math>.
  
the intersection of AB and OC, AD=BD=3.
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Using the Pythagorean Theorem again, <math>AC= \sqrt{AD^2 + CD^2} = \sqrt{3^2+1^2}=\sqrt{10} \Longrightarrow \textbf{(A)}</math>.
  
Also, OD=4. This means that CD=1.
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<center><asy>
 
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pen d = linewidth(0.7); pathpen = d; pointpen = black; pen f = fontsize(9);
Using the pythagorean theorem,  
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path p = CR((0,0),5);
 
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pair O = (0,0), A=(5,0), B = IP(p,CR(A,6)), C = IP(p,CR(A,3)), D=IP(A--B,O--C);
<math>AC=\sqrt{3^2+1^2}=\sqrt{10}</math>.
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D(p); D(MP("A",A,E)--D(MP("O",O))--MP("B",B,NE)--cycle); D(A--MP("C",C,ENE),dashed+d); D(O--C,dashed+d); D(rightanglemark(O,D(MP("D",D,W)),A)); MP("5",(A+O)/2); MP("3",(A+D)/2,SW);
 
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</asy></center>
Answer A is the correct answer.
 
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=B|num-b=9|num-a=11}}
 
{{AMC10 box|year=2008|ab=B|num-b=9|num-a=11}}
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[[Category:Introductory Geometry Problems]]

Revision as of 13:20, 11 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 $AC$?

$\mathrm{(A)}\ \sqrt{10}\qquad\mathrm{(B)}\ \frac{7}{2}\qquad\mathrm{(C)}\ \sqrt{14}\qquad\mathrm{(D)}\ \sqrt{15}\qquad\mathrm{(E)}\ 4$

Solution

Let the center of the circle be $O$, and let $D$ be the intersection of $\overline{AB}$ and $\overline{OC}$ (then $D$ is the midpoint of $\overline{AB}$). $OA=OB=5$, since they are both radii.

By the Pythagorean Theorem, $OD = \sqrt{OA^2 - DA^2} = 4$, and by subtraction, $CD=OC - OD = 1$.

Using the Pythagorean Theorem again, $AC= \sqrt{AD^2 + CD^2} = \sqrt{3^2+1^2}=\sqrt{10} \Longrightarrow \textbf{(A)}$.

[asy] pen d = linewidth(0.7); pathpen = d; pointpen = black; pen f = fontsize(9); path p = CR((0,0),5); pair O = (0,0), A=(5,0), B = IP(p,CR(A,6)), C = IP(p,CR(A,3)), D=IP(A--B,O--C); D(p); D(MP("A",A,E)--D(MP("O",O))--MP("B",B,NE)--cycle); D(A--MP("C",C,ENE),dashed+d); D(O--C,dashed+d); D(rightanglemark(O,D(MP("D",D,W)),A)); MP("5",(A+O)/2); MP("3",(A+D)/2,SW); [/asy]

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