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Difference between revisions of "2007 AMC 10B Problems/Problem 11"

m (boxed answers)
(→‎Solution 5)
 
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==Problem==
 
==Problem==
  
−
A circle passes through the three vertices of an isosceles triangle that has two sides of length <math>3</math> and a base of length <math>2.</math> What is the area of this circle?
+
A circle passes through the three vertices of an isosceles triangle that has two sides of length <math>3</math> and a base of length <math>2</math>. What is the area of this circle?
  
 
<math>\textbf{(A) } 2\pi \qquad\textbf{(B) } \frac{5}{2}\pi \qquad\textbf{(C) } \frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi</math>
 
<math>\textbf{(A) } 2\pi \qquad\textbf{(B) } \frac{5}{2}\pi \qquad\textbf{(C) } \frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi</math>
  
−
== Solution ==
+
== Solutions ==
 
=== Solution 1 ===
 
=== Solution 1 ===
−
Let <math>\triangle ABC</math> have vertex <math>A</math> and center <math>O</math>, with foot of altitude from <math>A</math> at <math>D</math>.  
+
Let <math>\triangle ABC</math> have vertex <math>A</math> and center <math>O</math>, with foot of altitude from <math>A</math> intersecting <math>BC</math> at <math>D</math>.  
 
<center><asy>
 
<center><asy>
 
import olympiad;
 
import olympiad;
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label("\(1\)",(B+D)/2,N);
 
label("\(1\)",(B+D)/2,N);
 
</asy></center>
 
</asy></center>
−
Then by Pythagorean Theorem (with radius <math>r</math>, height <math>OD = h</math>) on <math>\triangle OBD, ABD</math>
+
Then by the Pythagorean Theorem (with radius <math>r</math> and height <math>OD = h</math>) on <math>\triangle OBD</math> and <math>\triangle ABD</math>
 
<cmath>
 
<cmath>
 
\begin{align*} h^2 + 1 & = r^2 \\
 
\begin{align*} h^2 + 1 & = r^2 \\
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2R = \frac {AC}{\sin \angle ABC} = \frac {3}{\frac {2\sqrt {2}}{3}} \Longrightarrow R = \frac {9}{4\sqrt {2}}
 
2R = \frac {AC}{\sin \angle ABC} = \frac {3}{\frac {2\sqrt {2}}{3}} \Longrightarrow R = \frac {9}{4\sqrt {2}}
 
</cmath>
 
</cmath>
−
Answer follows as above.  
+
Answer follows as above.
 +
 
 +
=== Solution 3 ===
 +
 
 +
Extend segment <math>AD</math> to <math>R</math> on Circle <math>O</math>.
 +
<center><asy>
 +
import olympiad;
 +
pair B=(0,0), C=(2,0), A=(1,3), D=(1,0), R=(1,-0.35);
 +
pair O=circumcenter(A,B,C);
 +
draw(A--B--C--A--D--R--C);
 +
draw(B--O--C);
 +
draw(circumcircle(A,B,C));
 +
dot(O);
 +
label("\(A\)",A,N);
 +
label("\(B\)",B,S);
 +
label("\(C\)",C,S);
 +
label("\(D\)",D,S);
 +
label("\(O\)",O,W);
 +
label("\(R\)",R,S);
 +
label("\(r\)",(O+A)/2,SE);
 +
label("\(r\)",(O+R)/2,SE);
 +
label("\(3\)",(A+C)/2,NE);
 +
label("\(1\)",(C+D)/2,N);
 +
</asy></center>
 +
 
 +
By the Pythagorean Theorem <cmath>AD^2 = 3^2 - 1^2</cmath> <cmath>AD = 2\sqrt{2}</cmath>
 +
 
 +
<math>\triangle ADC</math> is similar to <math>\triangle ACR</math>, so <cmath>\frac {2\sqrt{2}}{3} = \frac {3}{2r}</cmath> which gives us <cmath>2r = \frac {9}{2\sqrt{2}} = \frac {9\sqrt{2}}{4}</cmath> therefore <cmath>r = \frac {9\sqrt{2}}{8}</cmath>
 +
 
 +
The area of the circle is therefore <math>\pi r^2 = \left(\frac {9\sqrt{2}}{8}\right)^2 \pi = \boxed{\mathrm{(C) \ } \frac {81}{32} \pi}</math>
 +
 
 +
===Solution 4===
 +
First, we extend <math>AD</math> to intersect the circle at <math>E.</math>
 +
<center><asy>
 +
import olympiad;
 +
pair B=(0,0), C=(2,0), A=(1,3), D=(1,0), E=(1,-(8^0.5)/8);
 +
pair O=circumcenter(A,B,C);
 +
draw(A--B--C--A--E);
 +
draw(circumcircle(A,B,C));
 +
dot(O);
 +
dot(D);
 +
dot(B);
 +
dot(C);
 +
dot(A);
 +
dot(E);
 +
label("$A$",A,N);
 +
label("$B$",B,S); label("\(C\)",C,S);
 +
label("$D$",D,NE);
 +
label("$O$",O,W);
 +
label("$E$",E,S);
 +
label("$3$",(A+B)/2,NW);
 +
label("$1$",(B+D)/2,N);
 +
</asy></center>
 +
By the [[Pythagorean Theorem]], we know that
 +
<cmath>1^2+AD^2=3^2\implies AD=2\sqrt{2}.</cmath>
 +
We also know that, from the [[Power of a Point Theorem]], <cmath>AD\cdot DE=BD\cdot DC.</cmath>
 +
We can substitute the values we know to get
 +
<cmath>2\sqrt{2}\cdot DE=1</cmath>
 +
We can simplify this to get that
 +
<cmath>DE=\dfrac{2\sqrt{2}}{8}.</cmath>
 +
We add <math>AD</math> and <math>DE</math> together to get the length of the diameter, and then we can find the area.
 +
<cmath>AE=AD+DE=2\sqrt{2}+\dfrac{2\sqrt{2}}{8}=\dfrac{9\sqrt{2}}{4}.</cmath>
 +
Therefore, the radius is <math>\dfrac{9\sqrt{2}}{8}</math>, so the area is
 +
<cmath>\left(\dfrac{9\sqrt{2}}{8}\right)^2\pi=\boxed{(\text{C})\dfrac{81}{32}\pi.}</cmath>
 +
 
 +
===Solution 5===
 +
Another possible solution is to plot the circle and triangle on a graph with the circle having center <math>(0,0)</math>.
 +
Let the radius of the circle = <math>r</math>.
 +
Let the distance between the origin and the base of triangle = <math>a</math>.
 +
<cmath>1 + a^2 = r^2</cmath>
 +
<cmath> r + a = 2\sqrt{2} </cmath>
 +
<cmath> a = 2\sqrt{2} - r </cmath>
 +
<cmath>9 - 4r\sqrt{2} = 0 </cmath>
 +
<cmath>r = \frac{9\sqrt{2}}{8}</cmath>
 +
<cmath>\pi r^2 = \boxed{(\text{C})\frac{81}{32}\pi}</cmath>
 +
 
 +
===Solution 6===
 +
We can also use LOC to solve this problem. If O is the center of the circle, angle <math>OBC</math> is double angle <math>BAC</math>. From law of cosines, we have that <math>\cos(\angle BAC) = \frac{7}{9}</math>. We now apply law of cosines on triangle <math>OBC</math> to get that <math>4 = r^2+r^2-2r^2\cos(2 \angle BAC)</math> but we know that <math>2 \cos (\angle BAC) = 2 \cdot (\frac{7}{9})^2 - 1 = \frac{17}{81}</math>. Plugging this in and simplifying, we get <math>r = \frac{9}{8} \cdot \sqrt{2}</math> and thus the area of the circle is equal to <math>\frac{81}{32} \pi</math>.
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 21:02, 26 July 2021

Problem

A circle passes through the three vertices of an isosceles triangle that has two sides of length $3$ and a base of length $2$. What is the area of this circle?

$\textbf{(A) } 2\pi \qquad\textbf{(B) } \frac{5}{2}\pi \qquad\textbf{(C) } \frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi$

Solutions

Solution 1

Let $\triangle ABC$ have vertex $A$ and center $O$, with foot of altitude from $A$ intersecting $BC$ at $D$.

[asy] import olympiad; pair B=(0,0), C=(2,0), A=(1,3), D=(1,0); pair O=circumcenter(A,B,C); draw(A--B--C--A--D); draw(B--O--C); draw(circumcircle(A,B,C)); dot(O); label("\(A\)",A,N); label("\(B\)",B,S); label("\(C\)",C,S); label("\(D\)",D,S); label("\(O\)",O,W); label("\(r\)",(O+A)/2,SE); label("\(r\)",(O+B)/2,N); label("\(h\)",(O+D)/2,SE); label("\(3\)",(A+B)/2,NW); label("\(1\)",(B+D)/2,N); [/asy]

Then by the Pythagorean Theorem (with radius $r$ and height $OD = h$) on $\triangle OBD$ and $\triangle ABD$ \begin{align*} h^2 + 1 & = r^2 \\ (h + r)^2 + 1 & = 9 \end{align*}

Substituting and solving gives $r = \frac {9}{4\sqrt {2}}$. Then the area of the circle is $r^2 \pi = \left(\frac {9}{4\sqrt {2}}\right)^2 \pi = \boxed{\mathrm{(C) \ } \frac {81}{32} \pi}$.

Solution 2

By $A = \frac {1}{2}Bh = \frac {abc}{4R}$ (or we could use $s = 4$ and Heron's formula), \[R = \frac {abc}{2Bh} = \frac {3 \cdot 3 \cdot 2}{2(2)(2\sqrt {2})} = \frac {9}{4\sqrt {2}}\] and the answer is $R^2 \pi = \boxed{\mathrm{(C) \ } \frac {81}{32} \pi}$

Alternatively, by the Extended Law of Sines, \[2R = \frac {AC}{\sin \angle ABC} = \frac {3}{\frac {2\sqrt {2}}{3}} \Longrightarrow R = \frac {9}{4\sqrt {2}}\] Answer follows as above.

Solution 3

Extend segment $AD$ to $R$ on Circle $O$.

[asy] import olympiad; pair B=(0,0), C=(2,0), A=(1,3), D=(1,0), R=(1,-0.35); pair O=circumcenter(A,B,C); draw(A--B--C--A--D--R--C); draw(B--O--C); draw(circumcircle(A,B,C)); dot(O); label("\(A\)",A,N); label("\(B\)",B,S); label("\(C\)",C,S); label("\(D\)",D,S); label("\(O\)",O,W); label("\(R\)",R,S); label("\(r\)",(O+A)/2,SE); label("\(r\)",(O+R)/2,SE); label("\(3\)",(A+C)/2,NE); label("\(1\)",(C+D)/2,N); [/asy]

By the Pythagorean Theorem \[AD^2 = 3^2 - 1^2\] \[AD = 2\sqrt{2}\]

$\triangle ADC$ is similar to $\triangle ACR$, so \[\frac {2\sqrt{2}}{3} = \frac {3}{2r}\] which gives us \[2r = \frac {9}{2\sqrt{2}} = \frac {9\sqrt{2}}{4}\] therefore \[r = \frac {9\sqrt{2}}{8}\]

The area of the circle is therefore $\pi r^2 = \left(\frac {9\sqrt{2}}{8}\right)^2 \pi = \boxed{\mathrm{(C) \ } \frac {81}{32} \pi}$

Solution 4

First, we extend $AD$ to intersect the circle at $E.$

[asy] import olympiad; pair B=(0,0), C=(2,0), A=(1,3), D=(1,0), E=(1,-(8^0.5)/8); pair O=circumcenter(A,B,C); draw(A--B--C--A--E); draw(circumcircle(A,B,C)); dot(O); dot(D); dot(B); dot(C); dot(A); dot(E); label("$A$",A,N); label("$B$",B,S); label("\(C\)",C,S); label("$D$",D,NE); label("$O$",O,W); label("$E$",E,S); label("$3$",(A+B)/2,NW); label("$1$",(B+D)/2,N); [/asy]

By the Pythagorean Theorem, we know that \[1^2+AD^2=3^2\implies AD=2\sqrt{2}.\] We also know that, from the Power of a Point Theorem, \[AD\cdot DE=BD\cdot DC.\] We can substitute the values we know to get \[2\sqrt{2}\cdot DE=1\] We can simplify this to get that \[DE=\dfrac{2\sqrt{2}}{8}.\] We add $AD$ and $DE$ together to get the length of the diameter, and then we can find the area. \[AE=AD+DE=2\sqrt{2}+\dfrac{2\sqrt{2}}{8}=\dfrac{9\sqrt{2}}{4}.\] Therefore, the radius is $\dfrac{9\sqrt{2}}{8}$, so the area is \[\left(\dfrac{9\sqrt{2}}{8}\right)^2\pi=\boxed{(\text{C})\dfrac{81}{32}\pi.}\]

Solution 5

Another possible solution is to plot the circle and triangle on a graph with the circle having center $(0,0)$. Let the radius of the circle = $r$. Let the distance between the origin and the base of triangle = $a$. \[1 + a^2 = r^2\] \[r + a = 2\sqrt{2}\] \[a = 2\sqrt{2} - r\] \[9 - 4r\sqrt{2} = 0\] \[r = \frac{9\sqrt{2}}{8}\] \[\pi r^2 = \boxed{(\text{C})\frac{81}{32}\pi}\]

Solution 6

We can also use LOC to solve this problem. If O is the center of the circle, angle $OBC$ is double angle $BAC$. From law of cosines, we have that $\cos(\angle BAC) = \frac{7}{9}$. We now apply law of cosines on triangle $OBC$ to get that $4 = r^2+r^2-2r^2\cos(2 \angle BAC)$ but we know that $2 \cos (\angle BAC) = 2 \cdot (\frac{7}{9})^2 - 1 = \frac{17}{81}$. Plugging this in and simplifying, we get $r = \frac{9}{8} \cdot \sqrt{2}$ and thus the area of the circle is equal to $\frac{81}{32} \pi$.

See also

2007 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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All AMC 10 Problems and Solutions

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