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Difference between revisions of "2011 AMC 12A Problems/Problem 24"

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\textbf{(E)}\ 2\sqrt{7} </math>
 
\textbf{(E)}\ 2\sqrt{7} </math>
  
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== Solution ==
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== Solution 1 ==
  
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=== Solution 1 ===
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Note as above that ABCD must be tangential to obtain the circle with maximal radius. Let <math>E</math>, <math>F</math>, <math>G</math>, and <math>H</math> be the points on <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math> respectively where the circle is tangent. Let <math>\theta=\angle BAD</math> and <math>\alpha=\angle ADC</math>. Since the quadrilateral is cyclic(because we want to maximize the circle, so we set the quadrilateral to be cyclic), <math>\angle ABC=180^{\circ}-\alpha</math> and <math>\angle BCD=180^{\circ}-\theta</math>. Let the circle have center <math>O</math> and radius <math>r</math>. Note that <math>OHD</math>, <math>OGC</math>, <math>OFB</math>, and <math>OEA</math> are right angles.  
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Note as above that ABCD must be tangential to obtain the circle with maximal radius. Let <math>E</math>, <math>F</math>, <math>G</math>, and <math>H</math> be the points on <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math> respectively where the circle is tangent. Let <math>\theta=\angle BAD</math> and <math>\alpha=\angle ADC</math>. Since the quadrilateral is cyclic, <math>\angle ABC=180^{\circ}-\alpha</math> and <math>\angle BCD=180^{\circ}-\theta</math>. Let the circle have center <math>O</math> and radius <math>r</math>. Note that <math>OHD</math>, <math>OGC</math>, <math>OFB</math>, and <math>OEA</math> are right angles.  
 
  
 
Hence <math>FOG=\theta</math>, <math>GOH=180^{\circ}-\alpha</math>, <math>EOH=180^{\circ}-\theta</math>, and <math>FOE=\alpha</math>.  
 
Hence <math>FOG=\theta</math>, <math>GOH=180^{\circ}-\alpha</math>, <math>EOH=180^{\circ}-\theta</math>, and <math>FOE=\alpha</math>.  
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Therefore, <math>AEOH\sim OFCG</math> and <math>EBFO\sim HOGD</math>.  
 
Therefore, <math>AEOH\sim OFCG</math> and <math>EBFO\sim HOGD</math>.  
  
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Let <math>x=CG</math>. Then <math>CF=x</math>, <math>BF=BE=9-x</math>, <math>GD=DH=7-x</math>, and <math>AH=AE=x+5</math>. Using <math>AEOH\sim OFCG</math> and <math>EBFO\sim HOGD</math> we have <math>r/(x+5)=x/r</math>, and <math>(9-x)/r=r/(7-x)</math>. By equating the value of <math>r^2</math> from each, <math>x(x+5)=(7-x)(9-x)</math>. Solving we obtain <math>x=3</math> so that <math>\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
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Let <math>x=CG</math>. Then <math>CF=x</math>, <math>BF=BE=9-x</math>, <math>GD=DH=7-x</math>, and <math>AH=AE=x+5</math>. Using <math>AEOH\sim OFCG</math> and <math>EBFO\sim HOGD</math> we have <math>r/(x+5)=x/r</math>, and <math>(9-x)/r=r/(7-x)</math>. By equating the value of <math>r^2</math> from each, <math>x(x+5)=(7-x)(9-x)</math>. Solving we obtain <math>x=3</math> so that <math>r=\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
  
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=== Solution 2 ===
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== Solution 2 ==
  
 
To maximize the radius of the circle, we also need to maximize its area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). In a tangential quadrilateral, the sum of opposite sides is equal to the semiperimeter of the quadrilateral. <math>14+7=12+9</math>, so this particular quadrilateral has an incircle. By definition, given <math>4</math> side lengths, a cyclic quadrilateral has the maximum area of any quadrilateral with those side lengths. Therefore, to maximize the area of the quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic.  
 
To maximize the radius of the circle, we also need to maximize its area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). In a tangential quadrilateral, the sum of opposite sides is equal to the semiperimeter of the quadrilateral. <math>14+7=12+9</math>, so this particular quadrilateral has an incircle. By definition, given <math>4</math> side lengths, a cyclic quadrilateral has the maximum area of any quadrilateral with those side lengths. Therefore, to maximize the area of the quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic.  
  
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For cyclic quadrilaterals, Brahmagupta's formula gives the area as <math>\sqrt{(s-a)(s-b)(s-c)(s-d)}</math> where <math>s</math> is the semiperimeter and <math>a, b, c,</math> and <math>d</math> are the side lengths. Breaking it up into <math>4</math> triangles, we see the area of a tangential quadrilateral is also equal to <math>r*s</math>. Equate these two equations.
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For cyclic quadrilaterals, [[Brahmagupta's formula]] gives the area as <math>\sqrt{(s-a)(s-b)(s-c)(s-d)}</math> where <math>s</math> is the semiperimeter and <math>a, b, c,</math> and <math>d</math> are the side lengths. Breaking it up into <math>4</math> triangles, we see the area of a tangential quadrilateral is also equal to <math>r*s</math>. Equate these two equations.
 
Substituting <math>s</math>, the semiperimeter, and <math>A</math>, the area and solving for <math>r</math>,we get <math>\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
 
Substituting <math>s</math>, the semiperimeter, and <math>A</math>, the area and solving for <math>r</math>,we get <math>\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
  
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=== Solution 3 (Trigonometry) ===
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== Solution 3 (Trigonometry) ==
  
 
By Pitot's Theorem, since <math>AB+CD=AD+BC</math>, there exists a circle tangent to all four sides of quadrilateral <math>ABCD</math>. Thus, we need to find the radius of this circle.
 
By Pitot's Theorem, since <math>AB+CD=AD+BC</math>, there exists a circle tangent to all four sides of quadrilateral <math>ABCD</math>. Thus, we need to find the radius of this circle.
  
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Let the circle be tangent to <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>AD</math> at points <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math>, respectively. Also, let <math>AP=x</math>. Then <math>AS</math> also equals <math>x</math>. Let the center of the circle be <math>O</math>. Observe that <math>AO</math> bisects angle <math>\angle A</math>, so <math>\cot\frac{A}{2}=\frac{x}{r}</math>. Moreover, <math>\cot\frac{C}{2}=\frac{9-(14-x)}{r}=\frac{x-5}{r}</math>. But <math>\angle \frac{D}{2}=90^\circ-\angle \frac{A}{2}</math>, so <math>\cot\frac{C}{2}=\tan\frac{A}{2}</math>, and we find that <math>\frac{r}{x}=\frac{x-5}{r}</math>. Hence, <math>r^2=x(x-5)</math>.
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Let the circle be tangent to <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>AD</math> at points <math>P</math>, <math>Q</math>, <math>R</math>, and <math>S</math>, respectively. Also, let <math>AP=x</math>. Then <math>AS</math> also equals <math>x</math>. Let the center of the circle be <math>O</math>. Observe that <math>AO</math> bisects angle <math>\angle A</math>, so <math>\cot\frac{A}{2}=\frac{x}{r}</math>. Moreover, <math>\cot\frac{C}{2}=\frac{9-(14-x)}{r}=\frac{x-5}{r}</math>. But <math>\angle \frac{C}{2}=90^\circ-\angle \frac{A}{2}</math>, so <math>\cot\frac{C}{2}=\tan\frac{A}{2}</math>, and we find that <math>\frac{r}{x}=\frac{x-5}{r}</math>. Hence, <math>r^2=x(x-5)</math>.
  
 
Similarly, <math>\cot\frac{B}{2}=\frac{14-x}{r}</math>, and <math>\cot\frac{D}{2}=\frac{12-x}{r}=\tan\frac{B}{2}=\frac{r}{14-x}</math>. Therefore, <math>r^2=\frac{12-x}{14-x}</math>. Putting this together with the above equation yields <cmath>(12-x)(14-x)=x(x-5)</cmath><cmath>\Longrightarrow x^2-26x+168=x^2-5x</cmath><cmath>\Longrightarrow 21x=168</cmath><cmath>\Longrightarrow x=8.</cmath> Thus, <math>r^2=8(8-5)=8(3)=24</math>, and <math>r=\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
 
Similarly, <math>\cot\frac{B}{2}=\frac{14-x}{r}</math>, and <math>\cot\frac{D}{2}=\frac{12-x}{r}=\tan\frac{B}{2}=\frac{r}{14-x}</math>. Therefore, <math>r^2=\frac{12-x}{14-x}</math>. Putting this together with the above equation yields <cmath>(12-x)(14-x)=x(x-5)</cmath><cmath>\Longrightarrow x^2-26x+168=x^2-5x</cmath><cmath>\Longrightarrow 21x=168</cmath><cmath>\Longrightarrow x=8.</cmath> Thus, <math>r^2=8(8-5)=8(3)=24</math>, and <math>r=\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
  
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=== Solution 4 (Areas) ===
 
  
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By Pitot's Theorem, since <math>AB+CD=AD+BC</math>, there exists a circle tangent to all four sides of quadrilateral <math>ABCD</math>. Let <math>E</math>, <math>F</math>, <math>G</math>, and <math>H</math> be the points on <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math> respectively where the circle is tangent. Let the inscribed circle have center <math>O</math> and radius <math>r</math>. Note that <math>OE</math>, <math>OF</math>, <math>OG</math>, and <math>OH</math> are perpendicular to sides <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math>, respectively. Thus <math>\angle AOH=\angle AOE</math>, <math>\angle BOE=\angle BOF</math>, <math>\angle COF=\angle COG</math>, and <math>\angle DOG=\angle DOH</math>.
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== Solution 4 (wowe) ==
 +
Let <math>r</math> be the inradius. Then, the area of the quadrilateral can be expressed as <math>21r</math>. Clearly, the larger the area of the quadrilateral, the larger the value of the inradius.
  
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From <math>\angle AOH+\angle AOE+\angle BOE+\angle BOF+\angle COF+\angle COG+\angle DOG+\angle DOH=360^{\circ}</math>, we get the two equations <math>\angle AOE+\angle BOE+\angle COG+\angle DOG=180^{\circ}</math> and <math>\angle AOH+\angle DOH+\angle BOF+\angle COF=180^{\circ}</math>. These are equivalent to <math>\angle AOB+\angle COD=180^{\circ}</math> and <math>\angle AOD+\angle BOC=180^{\circ}</math>.
 
  
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The ratio of the area of <math>AOB</math> to <math>COD</math> is <math>2</math>, since the heights are both <math>r</math>. However, we can also express this ratio as <math>\frac{\frac{1}{2}\times AO\times BO\times \sin (AOB)}{\frac{1}{2}\times CO\times DO\times \sin (COD)}</math>. Since <math>\angle AOB=180^{\circ}-\angle COD</math>, <math>\sin (AOB)=\sin (COD)</math>. Thus <math>\frac{AO\times BO}{CO\times DO}=2</math>. Similarly, <math>\frac{AO\times DO}{BO\times CO}=\frac{12}{9}=\frac{4}{3}</math>. From these two equations we have <math>BO=\frac{\sqrt{6}}{2}\times DO</math>.
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Know that the area of a quadrilateral is maximized when it is cyclic. Applying Brahmagupta's, the area will be <math>\sqrt{(9)(7)(12)(14)} = 42\sqrt{6}</math>. Since <math>21r = 42\sqrt{6}</math>, <math>r = \boxed{\textbf{(C) } 2\sqrt6}</math>.
  
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Now let <math>AH=AE=a</math>, <math>BE=BF=b</math>, <math>CF=CG=c</math>, and <math>DG=DH=d</math>. We thus have the equations <math>a+b=14</math>, <math>b+c=9</math>, <math>c+d=7</math>, and <math>d+a=12</math>. Solving gives us <math>a=8</math>, <math>b=6</math>, <math>c=3</math>, and <math>d=4</math>. By the Pythagorean theorem, we have <math>BO^2-b^2=DO^2-d^2</math>. This is <math>DO^2\times\frac{3}{2}-36=DO^2-16</math>, which gives us <math>DO^2=40</math>. Thus, <math>r=\sqrt{DO^2-d^2}=\sqrt{24}</math>, and <math>r=\boxed{\textbf{(C)}\ 2\sqrt{6}}</math>.
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-skibbysiggy
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2011|num-b=23|num-a=25|ab=A}}
 
{{AMC12 box|year=2011|num-b=23|num-a=25|ab=A}}
 +
 +
[[Category:Intermediate Geometry Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:49, 4 September 2024

Problem

Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?

$\textbf{(A)}\ \sqrt{15} \qquad \textbf{(B)}\ \sqrt{21} \qquad \textbf{(C)}\ 2\sqrt{6} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 2\sqrt{7}$

Solution 1

Note as above that ABCD must be tangential to obtain the circle with maximal radius. Let $E$, $F$, $G$, and $H$ be the points on $AB$, $BC$, $CD$, and $DA$ respectively where the circle is tangent. Let $\theta=\angle BAD$ and $\alpha=\angle ADC$. Since the quadrilateral is cyclic(because we want to maximize the circle, so we set the quadrilateral to be cyclic), $\angle ABC=180^{\circ}-\alpha$ and $\angle BCD=180^{\circ}-\theta$. Let the circle have center $O$ and radius $r$. Note that $OHD$, $OGC$, $OFB$, and $OEA$ are right angles.

Hence $FOG=\theta$, $GOH=180^{\circ}-\alpha$, $EOH=180^{\circ}-\theta$, and $FOE=\alpha$.

Therefore, $AEOH\sim OFCG$ and $EBFO\sim HOGD$.

Let $x=CG$. Then $CF=x$, $BF=BE=9-x$, $GD=DH=7-x$, and $AH=AE=x+5$. Using $AEOH\sim OFCG$ and $EBFO\sim HOGD$ we have $r/(x+5)=x/r$, and $(9-x)/r=r/(7-x)$. By equating the value of $r^2$ from each, $x(x+5)=(7-x)(9-x)$. Solving we obtain $x=3$ so that $r=\boxed{\textbf{(C)}\ 2\sqrt{6}}$.

Solution 2

To maximize the radius of the circle, we also need to maximize its area. To maximize the area of the circle, the quadrilateral must be tangential (have an incircle). In a tangential quadrilateral, the sum of opposite sides is equal to the semiperimeter of the quadrilateral. $14+7=12+9$, so this particular quadrilateral has an incircle. By definition, given $4$ side lengths, a cyclic quadrilateral has the maximum area of any quadrilateral with those side lengths. Therefore, to maximize the area of the quadrilateral and thus the incircle, we assume that this quadrilateral is cyclic.

For cyclic quadrilaterals, Brahmagupta's formula gives the area as $\sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter and $a, b, c,$ and $d$ are the side lengths. Breaking it up into $4$ triangles, we see the area of a tangential quadrilateral is also equal to $r*s$. Equate these two equations. Substituting $s$, the semiperimeter, and $A$, the area and solving for $r$,we get $\boxed{\textbf{(C)}\ 2\sqrt{6}}$.

Solution 3 (Trigonometry)

By Pitot's Theorem, since $AB+CD=AD+BC$, there exists a circle tangent to all four sides of quadrilateral $ABCD$. Thus, we need to find the radius of this circle.

Let the circle be tangent to $AB$, $BC$, $CD$, and $AD$ at points $P$, $Q$, $R$, and $S$, respectively. Also, let $AP=x$. Then $AS$ also equals $x$. Let the center of the circle be $O$. Observe that $AO$ bisects angle $\angle A$, so $\cot\frac{A}{2}=\frac{x}{r}$. Moreover, $\cot\frac{C}{2}=\frac{9-(14-x)}{r}=\frac{x-5}{r}$. But $\angle \frac{C}{2}=90^\circ-\angle \frac{A}{2}$, so $\cot\frac{C}{2}=\tan\frac{A}{2}$, and we find that $\frac{r}{x}=\frac{x-5}{r}$. Hence, $r^2=x(x-5)$.

Similarly, $\cot\frac{B}{2}=\frac{14-x}{r}$, and $\cot\frac{D}{2}=\frac{12-x}{r}=\tan\frac{B}{2}=\frac{r}{14-x}$. Therefore, $r^2=\frac{12-x}{14-x}$. Putting this together with the above equation yields \[(12-x)(14-x)=x(x-5)\]\[\Longrightarrow x^2-26x+168=x^2-5x\]\[\Longrightarrow 21x=168\]\[\Longrightarrow x=8.\] Thus, $r^2=8(8-5)=8(3)=24$, and $r=\boxed{\textbf{(C)}\ 2\sqrt{6}}$.


Solution 4 (wowe)

Let $r$ be the inradius. Then, the area of the quadrilateral can be expressed as $21r$. Clearly, the larger the area of the quadrilateral, the larger the value of the inradius.


Know that the area of a quadrilateral is maximized when it is cyclic. Applying Brahmagupta's, the area will be $\sqrt{(9)(7)(12)(14)} = 42\sqrt{6}$. Since $21r = 42\sqrt{6}$, $r = \boxed{\textbf{(C) } 2\sqrt6}$.

-skibbysiggy

See also

2011 AMC 12A (Problems • Answer Key • Resources)
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