Difference between revisions of "2006 AMC 10B Problems/Problem 19"
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A circle of radius <math>2</math> is centered at <math>O</math>. Square <math>OABC</math> has side length <math>1</math>. Sides <math>AB</math> and <math>CB</math> are extended past <math>B</math> to meet the circle at <math>D</math> and <math>E</math>, respectively. What is the area of the shaded region in the figure, which is bounded by <math>BD</math>, <math>BE</math>, and the minor arc connecting <math>D</math> and <math>E</math>? | A circle of radius <math>2</math> is centered at <math>O</math>. Square <math>OABC</math> has side length <math>1</math>. Sides <math>AB</math> and <math>CB</math> are extended past <math>B</math> to meet the circle at <math>D</math> and <math>E</math>, respectively. What is the area of the shaded region in the figure, which is bounded by <math>BD</math>, <math>BE</math>, and the minor arc connecting <math>D</math> and <math>E</math>? | ||
− | [[Image:2006amc10b19.gif]] | + | <!-- [[Image:2006amc10b19.gif]] --> |
− | + | <asy>defaultpen(linewidth(0.8)); | |
+ | pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B; | ||
+ | fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); | ||
+ | clip(B--Arc(O, 2, 30, 60)--cycle); | ||
+ | draw(Circle(origin, 2)); | ||
+ | draw((-2,0)--(2,0)^^(0,-2)--(0,2)); | ||
+ | draw(A--D^^C--E); | ||
+ | label("$A$", A, dir(point--A)); | ||
+ | label("$C$", C, dir(point--C)); | ||
+ | label("$O$", O, dir(point--O)); | ||
+ | label("$D$", D, dir(point--D)); | ||
+ | label("$E$", E, dir(point--E)); | ||
+ | label("$B$", B, SW);</asy> | ||
<math> \mathrm{(A) \ } \frac{\pi}{3}+1-\sqrt{3}\qquad \mathrm{(B) \ } \frac{\pi}{2}(2-\sqrt{3}) | <math> \mathrm{(A) \ } \frac{\pi}{3}+1-\sqrt{3}\qquad \mathrm{(B) \ } \frac{\pi}{2}(2-\sqrt{3}) | ||
\qquad \mathrm{(C) \ } \pi(2-\sqrt{3})\qquad \mathrm{(D) \ } \frac{\pi}{6}+\frac{\sqrt{3}+1}{2}\qquad \mathrm{(E) \ } \frac{\pi}{3}-1+\sqrt{3} </math> | \qquad \mathrm{(C) \ } \pi(2-\sqrt{3})\qquad \mathrm{(D) \ } \frac{\pi}{6}+\frac{\sqrt{3}+1}{2}\qquad \mathrm{(E) \ } \frac{\pi}{3}-1+\sqrt{3} </math> | ||
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The shaded area is equivalent to the area of sector <math>DOE</math>, minus the area of triangle <math>DOE</math> plus the area of triangle <math>DBE</math>. | The shaded area is equivalent to the area of sector <math>DOE</math>, minus the area of triangle <math>DOE</math> plus the area of triangle <math>DBE</math>. | ||
− | Using the Pythagorean Theorem | + | Using the [[Pythagorean Theorem]], <math>(DA)^2=(CE)^2=2^2-1^2=3</math> so <math>DA=CE=\sqrt{3}</math>. |
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− | <math>(DA)^2=(CE)^2=2^2-1^2=3</math> | ||
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− | <math>DA=CE=\sqrt{3} | ||
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− | Since <math>OABC</math> is a square, <math> \angle COA = 90^\circ </math>. | + | Clearly, <math>DOA</math> and <math>EOC</math> are <math>30-60-90</math> triangles with <math>\angle EOC = \angle DOA = 60^\circ </math>. Since <math>OABC</math> is a square, <math> \angle COA = 90^\circ </math>. |
<math>\angle DOE</math> can be found by doing some subtraction of angles. | <math>\angle DOE</math> can be found by doing some subtraction of angles. | ||
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The area of triangle <math>DOE</math> is <math> \frac{1}{2}\cdot 2 \cdot 2 \cdot \sin 30^\circ = 1 </math>. | The area of triangle <math>DOE</math> is <math> \frac{1}{2}\cdot 2 \cdot 2 \cdot \sin 30^\circ = 1 </math>. | ||
− | Since <math>AB=CB=1</math> , <math>DB=ED=(\sqrt{3}-1)</math>. | + | Since <math>AB=CB=1</math> , <math>DB=ED=(\sqrt{3}-1)</math>. So, the area of triangle <math>DBE</math> is <math>\frac{1}{2} \cdot (\sqrt{3}-1)^2 = 2-\sqrt{3}</math>. Therefore, the shaded area is <math> (\frac{\pi}{3}) - (1) + (2-\sqrt{3}) = \frac{\pi}{3}+1-\sqrt{3} \Longrightarrow \boxed{\mathrm{(A)}} </math> |
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− | So, the area of triangle <math>DBE</math> is <math>\frac{1}{2} \cdot (\sqrt{3}-1)^2 = 2-\sqrt{3}</math>. | ||
− | |||
− | Therefore, the shaded area is <math> (\frac{\pi}{3}) - (1) + (2-\sqrt{3}) = \frac{\pi}{3}+1-\sqrt{3} \ | ||
== See Also == | == See Also == | ||
− | + | {{AMC10 box|year=2006|ab=B|num-b=18|num-a=20}} | |
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[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
+ | [[Category:Area Problems]] | ||
+ | [[Category:Circle Problems]] |
Revision as of 23:28, 20 August 2011
Problem
A circle of radius is centered at . Square has side length . Sides and are extended past to meet the circle at and , respectively. What is the area of the shaded region in the figure, which is bounded by , , and the minor arc connecting and ?
Solution
The shaded area is equivalent to the area of sector , minus the area of triangle plus the area of triangle .
Using the Pythagorean Theorem, so .
Clearly, and are triangles with . Since is a square, .
can be found by doing some subtraction of angles.
So, the area of sector is .
The area of triangle is .
Since , . So, the area of triangle is . Therefore, the shaded area is
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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 |