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Difference between revisions of "2006 AMC 12A Problems/Problem 17"

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Extend <math>\overline{AD}</math> past <math>D</math> to meet the circle at <math>G \ne D</math>. Because <math>E</math> is collinear with <math>B</math> and <math>D</math>, <math>\angle EDG = 45^\circ.</math> Also, <math>ED = EG,</math> which implies <math>\angle EGD = 45^\circ</math>, so <math>\triangle EDG</math> is an isosceles right triangle. Thus <math>DG = r\sqrt{2}</math>. By the Power of a Point Theorem,
 
Extend <math>\overline{AD}</math> past <math>D</math> to meet the circle at <math>G \ne D</math>. Because <math>E</math> is collinear with <math>B</math> and <math>D</math>, <math>\angle EDG = 45^\circ.</math> Also, <math>ED = EG,</math> which implies <math>\angle EGD = 45^\circ</math>, so <math>\triangle EDG</math> is an isosceles right triangle. Thus <math>DG = r\sqrt{2}</math>. By the Power of a Point Theorem,
−
<cmath> 9+5\sqrt{2} = AF^2= AD\cdot AG = AD\cdot (AD+DG)=s(s+r\sqrt{2}) = s^2+rs\sqrt{2}.</cmath> As in the first solution, we conclude that <math>r/s=\boxed{5/9}</math>.
+
<cmath>\begin{align*}
 +
9+5\sqrt{2} &= AF^2 \\
 +
&= AD\cdot AG\\
 +
& = AD\cdot \left(AD+DG\right) \\
 +
&=
 +
s\left(s+r\sqrt{2}\right) \\
 +
&= s^2+rs\sqrt{2}.\end{align*}</cmath> As in the first solution, we conclude that <math>r/s=\boxed{5/9}</math>.
  
 
== See Also ==
 
== See Also ==

Revision as of 15:40, 22 August 2022

Problem

Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$. What is $r/s$?

AMC12 2006A 17.png

$\mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ } \frac{9}{5}$

Solutions

Solution 1

One possibility is to use the coordinate plane, setting $B$ at the origin. Point $A$ will be $(0,s)$ and $E$ will be $\left(s + \frac{r}{\sqrt{2}},\ s + \frac{r}{\sqrt{2}}\right)$ since $B, D$, and $E$ are collinear and contain a diagonal of $ABCD$. The Pythagorean theorem results in

\[AF^2 + EF^2 = AE^2\]

\[r^2 + \left(\sqrt{9 + 5\sqrt{2}}\right)^2 = \left(\left(s + \frac{r}{\sqrt{2}}\right) - 0\right)^2 + \left(\left(s + \frac{r}{\sqrt{2}}\right) - s\right)^2\]

\[r^2 + 9 + 5\sqrt{2} = s^2 + rs\sqrt{2} + \frac{r^2}{2} + \frac{r^2}{2}\]

\[9 + 5\sqrt{2} = s^2 + rs\sqrt{2}\]

This implies that $rs = 5$ and $s^2 = 9$; dividing gives us $\frac{r}{s} = \frac{5}{9} \Rightarrow B$.

Solution 2

First note that angle $\angle AFE$ is right since $\overline{AF}$ is tangent to the circle. Using the Pythagorean Theorem on $\triangle AFE$, then, we see \[AE^2 = 9 + 5\sqrt{2} + r^2.\]

But it can also be seen that $\angle BDA = 45^\circ$. Therefore, since $D$ lies on $\overline{BE}$, $\angle ADE = 135^\circ$. Using the Law of Cosines on $\triangle ADE$, we see

\[AE^2 = s^2 + r^2 - 2sr\cos(135^\circ)\] \[AE^2 = s^2 + r^2 - 2sr\left(-\frac{1}{\sqrt{2}}\right)\] \[AE^2 = s^2 + r^2 + \sqrt{2}sr\] \[9 + 5\sqrt{2} + r^2 = s^2 + r^2 + \sqrt{2}sr\] \[9 + 5\sqrt{2} = s^2 + \sqrt{2}sr.\]

Thus, since $r$ and $s$ are rational, $s^2 = 9$ and $sr = 5$. So $s = 3$, $r = \frac{5}{3}$, and $\frac{r}{s} = \frac{5}{9}$.

Solution 3

(Similar to Solution 1) First, draw line AE and mark a point Z that is equidistant from E and D so that $\angle{DZE} = 90^\circ$ and that line $\overline{AZ}$ includes point D. Since DE is equal to the radius $r$, $DZ=EZ=\frac{r}{\sqrt2}=\frac{r\sqrt2}{2}.$

Note that triangles $\triangle AFE$ and $\triangle AZE$ share the same hypotenuse $(AE)$, meaning that \[AZ^2+EZ^2=AF^2+EF^2\] Plugging in our values we have: \[\left(s+\frac{r\sqrt{2}}{2}\right)^2+\left(\frac{r\sqrt{2}}{2}\right)^2=\left(\sqrt{9+5\sqrt{2}}\right)^2+r^2\] \[s^2+sr\sqrt{2}+\frac{r^2}{2}+\frac{r^2}{2}=9+5\sqrt{2}+r^2\] \[s^2+sr\sqrt{2}=9+5\sqrt{2}\] By logic $s=3$ and $sr=5 \implies r=5/3.$

Therefore, $\frac{\frac{5}{3}}{3}=\frac{5}{9}=\boxed{B}$

Solution 4 - Alcumus

Let $B=(0,0)$, $C=(s,0)$, $A=(0,s)$, $D=(s,s)$, and $E=\left(s+\frac{r}{\sqrt{2}},s+\frac{r}{\sqrt{2}} \right)$. Apply the Pythagorean Theorem to $\triangle AFE$ to obtain \[r^2+\left(9+5\sqrt{2}\right)=\left(s+\frac{r}{\sqrt{2}}\right)^2+\left(\frac{r}{\sqrt{2}}\right)^2,\] from which $9+5\sqrt{2}=s^2+rs\sqrt{2}$. Because $r$ and $s$ are rational, it follows that $s^2=9$ and $rs=5$, so $r/s = \boxed{5/9}$.

OR

Extend $\overline{AD}$ past $D$ to meet the circle at $G \ne D$. Because $E$ is collinear with $B$ and $D$, $\angle EDG = 45^\circ.$ Also, $ED = EG,$ which implies $\angle EGD = 45^\circ$, so $\triangle EDG$ is an isosceles right triangle. Thus $DG = r\sqrt{2}$. By the Power of a Point Theorem, \begin{align*} 9+5\sqrt{2} &= AF^2 \\ &= AD\cdot AG\\ & = AD\cdot \left(AD+DG\right) \\ &= s\left(s+r\sqrt{2}\right) \\ &= s^2+rs\sqrt{2}.\end{align*} As in the first solution, we conclude that $r/s=\boxed{5/9}$.

See Also

2006 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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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