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Difference between revisions of "2017 AMC 10A Problems/Problem 22"

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<math>\textbf{(A) } \frac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}</math>
 
<math>\textbf{(A) } \frac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}</math>
  
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==Solution==
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==Solution 1==
 
<asy>
 
<asy>
 
real sqrt3 = 1.73205080757;
 
real sqrt3 = 1.73205080757;
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</asy>
 
</asy>
 
Let the radius of the circle be <math>r</math>, and let its center be <math>O</math>. Since <math>\overline{AB}</math> and <math>\overline{AC}</math> are tangent to circle <math>O</math>, then <math>\angle OBA = \angle OCA = 90^{\circ}</math>, so <math>\angle BOC = 120^{\circ}</math>. Therefore, since <math>\overline{OB}</math> and <math>\overline{OC}</math> are equal to <math>r</math>, then (pick your favorite method) <math>\overline{BC} = r\sqrt{3}</math>. The area of the equilateral triangle is <math>\frac{(r\sqrt{3})^2 \sqrt{3}}4 = \frac{3r^2 \sqrt{3}}4</math>, and the area of the sector we are subtracting from it is <math>\frac 13 \pi r^2 - \frac 12 \cdot r\sqrt{3} \cdot \frac{r}2 = \frac{\pi r^2}3 -\frac{r^2 \sqrt{3}}4</math>. The area outside of the circle is <math> \frac{3r^2 \sqrt{3}}4-\left(\frac{\pi r^2}3 -\frac{r^2 \sqrt{3}}4\right) = r^2 \sqrt{3} - \frac{\pi r^2}3</math>. Therefore, the answer is <cmath>\frac{r^2 \sqrt{3} - \frac{\pi r^2}3}{\frac{3r^2 \sqrt{3}}4} = \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt 3 \pi}{27}}</cmath>
 
Let the radius of the circle be <math>r</math>, and let its center be <math>O</math>. Since <math>\overline{AB}</math> and <math>\overline{AC}</math> are tangent to circle <math>O</math>, then <math>\angle OBA = \angle OCA = 90^{\circ}</math>, so <math>\angle BOC = 120^{\circ}</math>. Therefore, since <math>\overline{OB}</math> and <math>\overline{OC}</math> are equal to <math>r</math>, then (pick your favorite method) <math>\overline{BC} = r\sqrt{3}</math>. The area of the equilateral triangle is <math>\frac{(r\sqrt{3})^2 \sqrt{3}}4 = \frac{3r^2 \sqrt{3}}4</math>, and the area of the sector we are subtracting from it is <math>\frac 13 \pi r^2 - \frac 12 \cdot r\sqrt{3} \cdot \frac{r}2 = \frac{\pi r^2}3 -\frac{r^2 \sqrt{3}}4</math>. The area outside of the circle is <math> \frac{3r^2 \sqrt{3}}4-\left(\frac{\pi r^2}3 -\frac{r^2 \sqrt{3}}4\right) = r^2 \sqrt{3} - \frac{\pi r^2}3</math>. Therefore, the answer is <cmath>\frac{r^2 \sqrt{3} - \frac{\pi r^2}3}{\frac{3r^2 \sqrt{3}}4} = \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt 3 \pi}{27}}</cmath>
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==Solution 2==
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<asy>
 +
real sqrt3 = 1.73205080757;
 +
draw(Circle((4, 4), 4));
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draw((4-2*sqrt3,6)--(4,4)--(4+2*sqrt3,6)--(4-2*sqrt3,6)--(4,12)--(4+2*sqrt3,6));
 +
label("A", (4, 12.4));
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label("B", (-.3, 6.3));
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label("C", (8.3, 6.3));
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label("O", (4, 3.4));
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</asy>
 +
(same diagram as Solution 1)
 +
 +
WLOG, let the side length of the triangle be <math>1</math>. Then, the area of the triangle is <math>\frac{\sqrt{3}}{4}</math>. We are looking for <math>\frac{Area of the portion inside the triangle but outside the circle}{Area of triangle}</math>. Since <math>m\angle ABO = 90^{\circ}=</math>m\angle ACO = 90^{\circ}<math>, and </math>m\angle ABC = m\angle ACB = 60^{\circ}<math>, we know </math>m\angle OBC=m\angle OCB=30^{\circ}<math>, and </math>m\angle BOC = 120^{\circ}<math>. Drop an angle bisector of </math>O<math> onto </math>BC<math>, call the point of intersection </math>D<math>. By SAS congruence, </math>\triangle BDO \cong \triangle CDO<math>, by CPCTC (Congruent Parts of Congruent Triangles are Congruent) </math>BD \cong DC<math> and they both measure </math>frac{1}{2}<math>. By 30-60-90 triangle, </math>OC = BO = \frac{\sqrt{3}}{3}<math>. The area of the region bounded by arc BC is one-third the area of circle O, whose area is </math>\left(\frac{\sqrt{3}}{3}\right)^{2}\pi=\frac{1}{3}\pi<math>. Therefore, the area of the region bounded by arc BC is </math>\frac{1}{3}\cdot\frac{1}{3}\pi=\frac{\pi}{9}<math>. We are nearly there. By 30-60-90 triangle, we know </math>DO = \frac{\sqrt{3}}{6}<math>, so the area of </math>\triangle BOC<math> is </math>\frac{1\cdot\frac{\sqrt{3}}{6}}{2}=\frac{\sqrt{3}}{12}<math>. The area of the region inside both the triangle and circle is the area of the region bounded by arc BC minus the area of </math>\triangle BOC<math>: </math>\frac{\pi}{9}-\frac{\sqrt{3}}{12}<math>. The area of the region outside of the circle but inside the triangle is </math>\frac{\sqrt{3}}{4}-\left(\frac{\pi}{9}-\frac{\sqrt{3}}{12}\right)=\frac{\sqrt{3}}{4}-\frac{\pi}{9}+\frac{\sqrt{3}}{12}=\frac{\sqrt{3}}{3}-\frac{\pi}{9}<math> and the ratio is </math>\frac{\frac{\sqrt{3}}{3}-\frac{\pi}{9}}{\frac{\sqrt{3}}{4}}=\left(\frac{\sqrt{3}}{3}-\frac{\pi}{9}\right)\cdot\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3\sqrt{3}}-\frac{4\pi}{9\sqrt{3}}=\frac{4}{3}-\frac{4\sqrt{3}\pi}{27} \Longrightarrow \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt 3 \pi}{27}}$.
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 +
~JH. L
  
 
==Video Solution==
 
==Video Solution==

Revision as of 22:10, 13 June 2022

Problem

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?

$\textbf{(A) } \frac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\frac{4\sqrt{3}\pi}{27}$

Solution 1

[asy] real sqrt3 = 1.73205080757; draw(Circle((4, 4), 4)); draw((4-2*sqrt3,6)--(4,4)--(4+2*sqrt3,6)--(4-2*sqrt3,6)--(4,12)--(4+2*sqrt3,6)); label("A", (4, 12.4)); label("B", (-.3, 6.3)); label("C", (8.3, 6.3)); label("O", (4, 3.4)); [/asy] Let the radius of the circle be $r$, and let its center be $O$. Since $\overline{AB}$ and $\overline{AC}$ are tangent to circle $O$, then $\angle OBA = \angle OCA = 90^{\circ}$, so $\angle BOC = 120^{\circ}$. Therefore, since $\overline{OB}$ and $\overline{OC}$ are equal to $r$, then (pick your favorite method) $\overline{BC} = r\sqrt{3}$. The area of the equilateral triangle is $\frac{(r\sqrt{3})^2 \sqrt{3}}4 = \frac{3r^2 \sqrt{3}}4$, and the area of the sector we are subtracting from it is $\frac 13 \pi r^2 - \frac 12 \cdot r\sqrt{3} \cdot \frac{r}2 = \frac{\pi r^2}3 -\frac{r^2 \sqrt{3}}4$. The area outside of the circle is $\frac{3r^2 \sqrt{3}}4-\left(\frac{\pi r^2}3 -\frac{r^2 \sqrt{3}}4\right) = r^2 \sqrt{3} - \frac{\pi r^2}3$. Therefore, the answer is \[\frac{r^2 \sqrt{3} - \frac{\pi r^2}3}{\frac{3r^2 \sqrt{3}}4} = \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt 3 \pi}{27}}\]

Solution 2

[asy] real sqrt3 = 1.73205080757; draw(Circle((4, 4), 4)); draw((4-2*sqrt3,6)--(4,4)--(4+2*sqrt3,6)--(4-2*sqrt3,6)--(4,12)--(4+2*sqrt3,6)); label("A", (4, 12.4)); label("B", (-.3, 6.3)); label("C", (8.3, 6.3)); label("O", (4, 3.4)); [/asy] (same diagram as Solution 1)

WLOG, let the side length of the triangle be $1$. Then, the area of the triangle is $\frac{\sqrt{3}}{4}$. We are looking for $\frac{Area of the portion inside the triangle but outside the circle}{Area of triangle}$. Since $m\angle ABO = 90^{\circ}=$m\angle ACO = 90^{\circ}$, and$m\angle ABC = m\angle ACB = 60^{\circ}$, we know$m\angle OBC=m\angle OCB=30^{\circ}$, and$m\angle BOC = 120^{\circ}$. Drop an angle bisector of$O$onto$BC$, call the point of intersection$D$. By SAS congruence,$\triangle BDO \cong \triangle CDO$, by CPCTC (Congruent Parts of Congruent Triangles are Congruent)$BD \cong DC$and they both measure$frac{1}{2}$. By 30-60-90 triangle,$OC = BO = \frac{\sqrt{3}}{3}$. The area of the region bounded by arc BC is one-third the area of circle O, whose area is$\left(\frac{\sqrt{3}}{3}\right)^{2}\pi=\frac{1}{3}\pi$. Therefore, the area of the region bounded by arc BC is$\frac{1}{3}\cdot\frac{1}{3}\pi=\frac{\pi}{9}$. We are nearly there. By 30-60-90 triangle, we know$DO = \frac{\sqrt{3}}{6}$, so the area of$\triangle BOC$is$\frac{1\cdot\frac{\sqrt{3}}{6}}{2}=\frac{\sqrt{3}}{12}$. The area of the region inside both the triangle and circle is the area of the region bounded by arc BC minus the area of$\triangle BOC$:$\frac{\pi}{9}-\frac{\sqrt{3}}{12}$. The area of the region outside of the circle but inside the triangle is$\frac{\sqrt{3}}{4}-\left(\frac{\pi}{9}-\frac{\sqrt{3}}{12}\right)=\frac{\sqrt{3}}{4}-\frac{\pi}{9}+\frac{\sqrt{3}}{12}=\frac{\sqrt{3}}{3}-\frac{\pi}{9}$and the ratio is$\frac{\frac{\sqrt{3}}{3}-\frac{\pi}{9}}{\frac{\sqrt{3}}{4}}=\left(\frac{\sqrt{3}}{3}-\frac{\pi}{9}\right)\cdot\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3\sqrt{3}}-\frac{4\pi}{9\sqrt{3}}=\frac{4}{3}-\frac{4\sqrt{3}\pi}{27} \Longrightarrow \boxed{\textbf{(E) } \frac 43 - \frac{4\sqrt 3 \pi}{27}}$.

~JH. L

Video Solution

https://www.youtube.com/watch?v=GnJDNtjd57k&feature=youtu.be

https://youtu.be/ADDAOhNAsjQ -Video Solution by Richard Rusczyk

See Also

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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All AMC 10 Problems and Solutions

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