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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 9"

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(→‎Video Solution (Just 3 min!))
 
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Construct the circle that passes through <math>A</math>, <math>O</math>, and <math>C</math>, centered at <math>X</math>.  
 
Construct the circle that passes through <math>A</math>, <math>O</math>, and <math>C</math>, centered at <math>X</math>.  
  
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Then connect <math>OX</math>, and notice that <math>OX</math> is the perpendicular bisector of <math>AC</math>. Let the intersection of <math>OX</math> with <math>AC</math> be <math>D</math>.
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Also notice that <math>\overline{OA}</math> and <math>\overline{OC}</math> are the angle bisectors of angle <math>\angle BAC</math> and <math>\angle BCA</math> respectively. We then deduce <math>\angle AOC=120^\circ</math>.
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Also notice that <math>AO</math> and <math>CO</math> are the angle bisectors of angle <math>BAC</math> and <math>BCA</math> respectively. We then deduce <math>AOC=120^\circ</math>.
 
  
 
Consider another point <math>M</math> on Circle <math>X</math> opposite to point <math>O</math>.  
 
Consider another point <math>M</math> on Circle <math>X</math> opposite to point <math>O</math>.  
  
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As <math>AOCM</math> an inscribed quadrilateral of Circle <math>X</math>, <math>AMC=180^\circ-120^\circ=60^\circ</math>.
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As <math>AOCM</math> is an inscribed quadrilateral of Circle <math>X</math>, <math>\angle AMC=180^\circ-120^\circ=60^\circ</math>.
  
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Afterward, deduce that <math>AXC=2·AMC=120^\circ</math>.
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Afterward, deduce that <math>\angle AXC=2·\angle AMC=120^\circ</math>.
  
 
By the Cosine Rule, we have the equation: (where <math>r</math> is the radius of circle <math>X</math>)
 
By the Cosine Rule, we have the equation: (where <math>r</math> is the radius of circle <math>X</math>)
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<math>r^2=12</math>
 
<math>r^2=12</math>
  
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The area is therefore <math>12\pi \Rightarrow \boxed{\textbf{(B)}}</math>.
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The area is therefore <math> \pi r^2 = \boxed{\textbf{(B)}\ 12\pi}</math>.
  
 
~Wilhelm Z
 
~Wilhelm Z
 +
 +
== Solution 2 ==
 +
We have <math>\angle AOC = 120^\circ</math>.
 +
 +
Denote by <math>R</math> the circumradius of <math>\triangle AOC</math>.
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In <math>\triangle AOC</math>, the law of sines implies
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<cmath>
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\[
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2 R = \frac{AC}{\sin \angle AOC} = 4 \sqrt{3} .
 +
\]
 +
</cmath>
 +
 +
Hence, the area of the circumcircle of <math>\triangle AOC</math> is
 +
<cmath>
 +
\[
 +
\pi R^2 = 12 \pi .
 +
\]
 +
</cmath>
 +
 +
Therefore, the answer is <math>\boxed{\textbf{(B) }12 \pi}</math>.
 +
 +
~Steven Chen (www.professorchenedu.com)
 +
 +
==Solution 3 ==
 +
 +
As in the previous solution, construct the circle that passes through <math>A</math>, <math>O</math>, and <math>C</math>, centered at <math>X</math>. Let <math>Y</math> be the intersection of <math>\overline{OX}</math> and <math>\overline{AB}</math>.
 +
 +
Note that since <math>\overline{OA}</math> is the angle bisector of <math>\angle BAC</math> that <math>\angle OAC=30^\circ</math>. Also by symmetry, <math>\overline{OX}</math> <math>\perp</math> <math>\overline{AB}</math> and <math>AY = 3</math>. Thus <math>\tan(30^\circ) = \frac{OY}{3}</math> so <math>OY = \sqrt{3}</math>.
 +
 +
Let <math>r</math> be the radius of circle <math>X</math>, and note that <math>AX = OX = r</math>. So <math>\triangle AYX</math> is a right triangle with legs of length <math>3</math> and <math>r - \sqrt{3}</math> and hypotenuse <math>r</math>. By Pythagoras, <math>3^2 + (r - \sqrt{3})^2 = r^2</math>. So <math>r = 2\sqrt{3}</math>.
 +
 +
Thus the area is <math> \pi r^2 = \boxed{\textbf{(B)}\ 12\pi}</math>.
 +
 +
-SharpeMind
 +
 +
==Solution 5 (SIMPLE) ==
 +
 +
The semiperimeter is <math>\frac{6+6+6}{2}=9</math> units.
 +
The area of the triangle is <math>9\sqrt{3}</math> units squared.  By the formula that says that the area of the triangle is its semiperimeter times its inradius, the inradius <math>r=\sqrt{3}</math>. As <math>\angle{AOC}=120^\circ</math>, we can form an altitude from point <math>O</math> to side <math>AC</math> at point <math>M</math>, forming two 30-60-90 triangles. As <math>CM=MA=3</math>, we can solve for <math>OC=2\sqrt{3}</math>. Now, the area of the circle is just <math>\pi*(2*\sqrt{3})^2 = 12\pi</math>. Select <math>\boxed{B}</math>.
 +
 +
~hastapasta, bob4108
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 +
==Solution 6 (Ptolemy) ==
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Call the diameter of the circle <math>d</math>. If we extend points <math>A</math> and <math>C</math> to meet at a point on the circle and call it <math>E</math>, then <math>\bigtriangleup OAE=\bigtriangleup OCE</math> . Note that both triangles are right, since their hypotenuse is the diameter of the circle. Therefore, <math>CE=AE=\sqrt{d^2-12}</math>. We know this since <math>OC=OA=OB</math> and <math>OC</math> is the hypotenuse of a <math>30-60-90</math> right triangle, with the longer leg being <math>\frac{6}{2}=3</math> so <math>OC=2\sqrt{3}</math>. Applying Ptolemy's Theorem on cyclic quadrilateral <math>OCEA</math>, we get <math>2({\sqrt{d^2-12}})\cdot{2\sqrt{3}}=6d</math>. Squaring and solving we get <math>d^2=48 \Longrightarrow (2r)^2=48</math> so <math>r^2=12</math>. Therefore, the area of the circle is <math>\boxed{12\pi}</math>
 +
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Magnetoninja Magnetoninja]
 +
 +
 +
==Video Solution (Just 3 min!)==
 +
https://youtu.be/kkm1d09bVpM
 +
 +
<i>~Education, the Study of Everything</i>
 +
 +
==Video Solution by TheBeautyofMath==
 +
https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1039
 +
 +
~IceMatrix
 +
 +
==See Also==
 +
{{AMC12 box|year=2021 Fall|ab=B|num-a=10|num-b=8}}
 +
 +
[[Category:Introductory Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 21:33, 10 July 2023

Problem

Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?

$\textbf{(A)} \: 9\pi \qquad\textbf{(B)} \: 12\pi \qquad\textbf{(C)} \: 18\pi \qquad\textbf{(D)} \: 24\pi \qquad\textbf{(E)} \: 27\pi$

Solution 1 (Cosine Rule)

Construct the circle that passes through $A$, $O$, and $C$, centered at $X$.

Also notice that $\overline{OA}$ and $\overline{OC}$ are the angle bisectors of angle $\angle BAC$ and $\angle BCA$ respectively. We then deduce $\angle AOC=120^\circ$.

Consider another point $M$ on Circle $X$ opposite to point $O$.

As $AOCM$ is an inscribed quadrilateral of Circle $X$, $\angle AMC=180^\circ-120^\circ=60^\circ$.

Afterward, deduce that $\angle AXC=2·\angle AMC=120^\circ$.

By the Cosine Rule, we have the equation: (where $r$ is the radius of circle $X$)

$2r^2(1-\cos(120^\circ))=6^2$

$r^2=12$

The area is therefore $\pi r^2 = \boxed{\textbf{(B)}\ 12\pi}$.

~Wilhelm Z

Solution 2

We have $\angle AOC = 120^\circ$.

Denote by $R$ the circumradius of $\triangle AOC$. In $\triangle AOC$, the law of sines implies \[ 2 R = \frac{AC}{\sin \angle AOC} = 4 \sqrt{3} . \]

Hence, the area of the circumcircle of $\triangle AOC$ is \[ \pi R^2 = 12 \pi . \]

Therefore, the answer is $\boxed{\textbf{(B) }12 \pi}$.

~Steven Chen (www.professorchenedu.com)

Solution 3

As in the previous solution, construct the circle that passes through $A$, $O$, and $C$, centered at $X$. Let $Y$ be the intersection of $\overline{OX}$ and $\overline{AB}$.

Note that since $\overline{OA}$ is the angle bisector of $\angle BAC$ that $\angle OAC=30^\circ$. Also by symmetry, $\overline{OX}$ $\perp$ $\overline{AB}$ and $AY = 3$. Thus $\tan(30^\circ) = \frac{OY}{3}$ so $OY = \sqrt{3}$.

Let $r$ be the radius of circle $X$, and note that $AX = OX = r$. So $\triangle AYX$ is a right triangle with legs of length $3$ and $r - \sqrt{3}$ and hypotenuse $r$. By Pythagoras, $3^2 + (r - \sqrt{3})^2 = r^2$. So $r = 2\sqrt{3}$.

Thus the area is $\pi r^2 = \boxed{\textbf{(B)}\ 12\pi}$.

-SharpeMind

Solution 5 (SIMPLE)

The semiperimeter is $\frac{6+6+6}{2}=9$ units. The area of the triangle is $9\sqrt{3}$ units squared. By the formula that says that the area of the triangle is its semiperimeter times its inradius, the inradius $r=\sqrt{3}$. As $\angle{AOC}=120^\circ$, we can form an altitude from point $O$ to side $AC$ at point $M$, forming two 30-60-90 triangles. As $CM=MA=3$, we can solve for $OC=2\sqrt{3}$. Now, the area of the circle is just $\pi*(2*\sqrt{3})^2 = 12\pi$. Select $\boxed{B}$.

~hastapasta, bob4108

Solution 6 (Ptolemy)

Call the diameter of the circle $d$. If we extend points $A$ and $C$ to meet at a point on the circle and call it $E$, then $\bigtriangleup OAE=\bigtriangleup OCE$ . Note that both triangles are right, since their hypotenuse is the diameter of the circle. Therefore, $CE=AE=\sqrt{d^2-12}$. We know this since $OC=OA=OB$ and $OC$ is the hypotenuse of a $30-60-90$ right triangle, with the longer leg being $\frac{6}{2}=3$ so $OC=2\sqrt{3}$. Applying Ptolemy's Theorem on cyclic quadrilateral $OCEA$, we get $2({\sqrt{d^2-12}})\cdot{2\sqrt{3}}=6d$. Squaring and solving we get $d^2=48 \Longrightarrow (2r)^2=48$ so $r^2=12$. Therefore, the area of the circle is $\boxed{12\pi}$

~Magnetoninja


Video Solution (Just 3 min!)

https://youtu.be/kkm1d09bVpM

~Education, the Study of Everything

Video Solution by TheBeautyofMath

https://www.youtube.com/watch?v=4qgYrCYG-qw&t=1039

~IceMatrix

See Also

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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 12 Problems and Solutions

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