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 <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>\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> |
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− | (D) | ||
==Solution== | ==Solution== | ||
− | Let the center of the circle be O | + | 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. |
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− | OA=OB=5, since they are both radii. | ||
− | + | By the [[Pythagorean Theorem]], <math>OD = \sqrt{OA^2 - DA^2} = 4</math>, and by subtraction, <math>CD=OC - OD = 1</math>. | |
− | the | + | Using the Pythagorean Theorem again, <math>AC= \sqrt{AD^2 + CD^2} = \sqrt{3^2+1^2}=\sqrt{10} \Longrightarrow \textbf{(A)}</math>. |
− | + | <center><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></center> | |
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==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 and are on a circle of radius and . Point is the midpoint of the minor arc . What is the length of the line segment ?
Solution
Let the center of the circle be , and let be the intersection of and (then is the midpoint of ). , since they are both radii.
By the Pythagorean Theorem, , and by subtraction, .
Using the Pythagorean Theorem again, .
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
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 |