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>? | ||
− | < | + | <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; | 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); | fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); | ||
Line 15: | Line 15: | ||
label("$D$", D, dir(point--D)); | label("$D$", D, dir(point--D)); | ||
label("$E$", E, dir(point--E)); | label("$E$", E, dir(point--E)); | ||
− | label("$B$", B, SW);</asy> | + | 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> | ||
− | == Solution == | + | == Solution 1 == |
− | The shaded area is equivalent to the area of sector <math>DOE</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]], <math>(DA)^2=(CE)^2=2^2-1^2=3</math> so <math>DA=CE=\sqrt{3}</math>. | Using the [[Pythagorean Theorem]], <math>(DA)^2=(CE)^2=2^2-1^2=3</math> so <math>DA=CE=\sqrt{3}</math>. | ||
Line 28: | Line 30: | ||
<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. | ||
− | <math> \angle COA - \angle DOA = \angle EOA </math> | + | <math> \angle COA - \angle DOA = \angle DOC = \angle EOA </math> |
<math> 90^\circ - 60^\circ = \angle EOA = 30^\circ </math> | <math> 90^\circ - 60^\circ = \angle EOA = 30^\circ </math> | ||
Line 40: | Line 42: | ||
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= | + | Since <math>AB=CB=1</math> , <math>DB=EB=(\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}) = \boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}</math> |
+ | |||
+ | OR | ||
+ | |||
+ | <math>\triangle{ODA}</math> has the same height as <math>\triangle{OBD}</math> which is <math>1.</math> | ||
+ | |||
+ | We already know that <math>BD = \sqrt{3} - 1.</math> | ||
+ | |||
+ | Therefore the area of <math>\triangle{OBD}</math> is <math>(\sqrt{3}-1) \cdot 1 \cdot \frac{1}{2} = \frac{\sqrt{3}-1}{2}.</math> | ||
+ | |||
+ | Since <math>\triangle{OBD} = \triangle{OBE} = \frac{\sqrt{3}-1}{2}.</math> | ||
+ | |||
+ | Therefore the sum of the areas is <math>2 \cdot \frac{\sqrt{3}-1}{2} = \sqrt{3}-1.</math> | ||
+ | |||
+ | Then the area of the shaded area becomes <math>\frac{\pi}{3} - (\sqrt{3} - 1) = \boxed{\textbf{(A) }\frac{\pi}{3} +1 - \sqrt{3}}.</math> | ||
+ | |||
+ | ~mathboy282 | ||
+ | |||
+ | == Solution 2 == | ||
+ | From the Pythagorean Theorem, we can see that <math>DA</math> is <math>\sqrt{3}</math>. Then, <math>DB = DA - BA = \sqrt{3} - 1</math>. The area of the shaded element is the area of sector <math>DOE</math> minus the areas of triangle <math>DBO</math> and triangle <math>EBO</math> combined. Below is an image to help. | ||
+ | |||
+ | <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^^D--O^^E--O^^B--O); | ||
+ | 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("$B$", (1.33,1.04), SW); | ||
+ | </asy> | ||
+ | |||
+ | Using the Base Altitude formula, where <math>DB</math> and <math>BE</math> are the bases and <math>OA</math> and <math>CO</math> are the altitudes, respectively, <math>[DBO] = [EBO] = \frac{\sqrt{3}-1}{2}</math>. The area of sector <math>DOE</math> is <math>\frac{1}{12}</math> of circle <math>O</math>. The area of circle <math>O</math> is <math>4\pi</math>, and therefore we have the area of sector <math>DBE</math> to be <math>\boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}</math>. | ||
+ | |||
+ | == Solution 3 (Using Answer Choices) == | ||
+ | Like the first solutions, you find that the area of sector <math>DOE</math> is <math>\frac{\pi}{3}</math>. We also know that the triangles will not be in terms of <math>{\pi}</math>. Looking at the answers, choices <math>\text{(A)}</math> and <math>\text{(E)}</math> both contain <math>\frac{\pi}{3}</math>. However, based on the diagram, we observe that the answer must be less than <math>\frac {\pi}{3}</math>. Only <math>\boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}</math> consists of a value less than <math>\frac{\pi}{3}</math>. | ||
+ | |||
+ | == Solution 4 == | ||
+ | |||
+ | We calculate the shaded area by subtracting the unshaded area in the quarter circle from a quarter circle, which consists of two <math> 30 ^{\circ} </math> sectors and two 30-60-90 triangles minus a square. The area of the shaded region is <math> \frac {\pi \times 2^2}{4} - 2 ( \frac {\pi \times 2^2}{12} + \frac {\sqrt 3}{2} - 1 ) = \boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}</math> | ||
+ | |||
+ | ~thatmathsguy | ||
== See Also == | == See Also == |
Latest revision as of 15:39, 29 May 2023
Contents
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 1
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
OR
has the same height as
which is
We already know that
Therefore the area of is
Since
Therefore the sum of the areas is
Then the area of the shaded area becomes
~mathboy282
Solution 2
From the Pythagorean Theorem, we can see that is
. Then,
. The area of the shaded element is the area of sector
minus the areas of triangle
and triangle
combined. Below is an image to help.
Using the Base Altitude formula, where and
are the bases and
and
are the altitudes, respectively,
. The area of sector
is
of circle
. The area of circle
is
, and therefore we have the area of sector
to be
.
Solution 3 (Using Answer Choices)
Like the first solutions, you find that the area of sector is
. We also know that the triangles will not be in terms of
. Looking at the answers, choices
and
both contain
. However, based on the diagram, we observe that the answer must be less than
. Only
consists of a value less than
.
Solution 4
We calculate the shaded area by subtracting the unshaded area in the quarter circle from a quarter circle, which consists of two sectors and two 30-60-90 triangles minus a square. The area of the shaded region is
~thatmathsguy
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.