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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]] -->
+
<asy>
−
<asy>defaultpen(linewidth(0.8));
+
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);
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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 1 ==
 
== Solution 1 ==
−
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]], <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>.
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<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=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>
+
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>
  
−
==Solution 2==
+
~thatmathsguy
−
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. 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>\frac{\pi}{3} + 1 - \sqrt{3} \Longrightarrow \boxed{A}</math>
 
  
 
== See Also ==
 
== See Also ==

Latest revision as of 14:39, 29 May 2023

Problem

A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$?

[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]

$\mathrm{(A) \ } \frac{\pi}{3}+1-\sqrt{3}\qquad \mathrm{(B) \ } \frac{\pi}{2}(2-\sqrt{3}) \qquad \mathrm{(C) \ } \pi(2-\sqrt{3})\qquad \mathrm{(D) \ } \frac{\pi}{6}+\frac{\sqrt{3}+1}{2}\qquad \mathrm{(E) \ } \frac{\pi}{3}-1+\sqrt{3}$

Solution 1

The shaded area is equivalent to the area of sector $DOE$ minus the area of triangle $DOE$ plus the area of triangle $DBE$.

Using the Pythagorean Theorem, $(DA)^2=(CE)^2=2^2-1^2=3$ so $DA=CE=\sqrt{3}$.

Clearly, $DOA$ and $EOC$ are $30-60-90$ triangles with $\angle EOC = \angle DOA = 60^\circ$. Since $OABC$ is a square, $\angle COA = 90^\circ$.

$\angle DOE$ can be found by doing some subtraction of angles.

$\angle COA - \angle DOA = \angle DOC = \angle EOA$

$90^\circ - 60^\circ = \angle EOA = 30^\circ$

$\angle DOA - \angle EOA = \angle DOE$

$60^\circ - 30^\circ = \angle DOE = 30^\circ$

So, the area of sector $DOE$ is $\frac{30}{360} \cdot \pi \cdot 2^2 = \frac{\pi}{3}$.

The area of triangle $DOE$ is $\frac{1}{2}\cdot 2 \cdot 2 \cdot \sin 30^\circ = 1$.

Since $AB=CB=1$ , $DB=EB=(\sqrt{3}-1)$. So, the area of triangle $DBE$ is $\frac{1}{2} \cdot (\sqrt{3}-1)^2 = 2-\sqrt{3}$. Therefore, the shaded area is $(\frac{\pi}{3}) - (1) + (2-\sqrt{3}) = \boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}$

OR

$\triangle{ODA}$ has the same height as $\triangle{OBD}$ which is $1.$

We already know that $BD = \sqrt{3} - 1.$

Therefore the area of $\triangle{OBD}$ is $(\sqrt{3}-1) \cdot 1 \cdot \frac{1}{2} = \frac{\sqrt{3}-1}{2}.$

Since $\triangle{OBD} = \triangle{OBE} = \frac{\sqrt{3}-1}{2}.$

Therefore the sum of the areas is $2 \cdot \frac{\sqrt{3}-1}{2} = \sqrt{3}-1.$

Then the area of the shaded area becomes $\frac{\pi}{3} - (\sqrt{3} - 1) = \boxed{\textbf{(A) }\frac{\pi}{3} +1 - \sqrt{3}}.$

~mathboy282

Solution 2

From the Pythagorean Theorem, we can see that $DA$ is $\sqrt{3}$. Then, $DB = DA - BA = \sqrt{3} - 1$. The area of the shaded element is the area of sector $DOE$ minus the areas of triangle $DBO$ and triangle $EBO$ 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 $DB$ and $BE$ are the bases and $OA$ and $CO$ are the altitudes, respectively, $[DBO] = [EBO] = \frac{\sqrt{3}-1}{2}$. The area of sector $DOE$ is $\frac{1}{12}$ of circle $O$. The area of circle $O$ is $4\pi$, and therefore we have the area of sector $DBE$ to be $\boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}$.

Solution 3 (Using Answer Choices)

Like the first solutions, you find that the area of sector $DOE$ is $\frac{\pi}{3}$. We also know that the triangles will not be in terms of ${\pi}$. Looking at the answers, choices $\text{(A)}$ and $\text{(E)}$ both contain $\frac{\pi}{3}$. However, based on the diagram, we observe that the answer must be less than $\frac {\pi}{3}$. Only $\boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}$ consists of a value less than $\frac{\pi}{3}$.

Solution 4

We calculate the shaded area by subtracting the unshaded area in the quarter circle from a quarter circle, which consists of two $30 ^{\circ}$ sectors and two 30-60-90 triangles minus a square. The area of the shaded region is $\frac {\pi \times 2^2}{4} - 2 ( \frac {\pi \times 2^2}{12} + \frac {\sqrt 3}{2} - 1 ) = \boxed{\textbf{(A) }\frac{\pi}{3} + 1 - \sqrt{3}}$

~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

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