Difference between revisions of "2011 AMC 10A Problems/Problem 18"
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== Problem 18 == | == Problem 18 == | ||
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Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math>C</math> has a point of tangency with the midpoint of <math>\overline{AB}</math>. What is the area inside Circle <math>C</math> but outside circle <math>A</math> and circle <math>B</math> ? | Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math>C</math> has a point of tangency with the midpoint of <math>\overline{AB}</math>. What is the area inside Circle <math>C</math> but outside circle <math>A</math> and circle <math>B</math> ? | ||
+ | <asy> pathpen = linewidth(.7); pointpen = black; pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1)); </asy> | ||
== Solution == | == Solution == |
Revision as of 20:49, 6 September 2011
Problem 18
Circles and each have radius 1. Circles and share one point of tangency. Circle has a point of tangency with the midpoint of . What is the area inside Circle but outside circle and circle ?
Solution
Draw a rectangle with vertices at the centers of and and the intersection of and . Then, we can compute the shaded area as the area of half of plus the area of the rectangle minus the area of the two sectors created by and . This is .
See Also
2011 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 |