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Difference between revisions of "2011 AMC 10B Problems/Problem 18"

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[[Category: Introductory Geometry Problems]]
 
[[Category: Introductory Geometry Problems]]
  
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==Solution==
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==Solution 1==
  
 
<center><asy>
 
<center><asy>
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x &= \boxed{\textbf{(E)} 75}
 
x &= \boxed{\textbf{(E)} 75}
 
\end{align*}</cmath>
 
\end{align*}</cmath>
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==Solution 2==
 +
 +
Let <math>\angle{DMC} = \angle{AMD} = \theta</math>. If we let <math>AM = x</math>, we have that <math>MD = \sqrt{x^2 + 9}</math>, by the Pythagorean Theorem, and similarily, <math>MC = \sqrt{x^2 - 12x + 45}</math>. Applying LOC, we see that
 +
<cmath>2x^2 + 54 - 12x - 2 \sqrt{x^4 - 12x^3 + 54x^2 - 108x + 405} \cdot cos (\theta) = 36</cmath> and <cmath>tan (\theta) = \frac{3}{x}</cmath>. YAY!!! We have two equations for two variables... that are terribly ugly. Well, we'll try to solve it. First of all, note that <math>sin (\theta) = \frac{3}{\sqrt{x^2+9}}</math>, so solving for <math>cos (\theta)</math> in terms of <math>x</math>, we get that <math>cos (\theta) = \frac{x \cdot \sqrt{x^2 + 9}}{x^2 + 9}</math>. The equation now becomes
 +
 +
<cmath>2x^2 + 54 - 12x - 2 \sqrt{x^4 - 12x^3 + 54x^2 - 108x + 405} \cdot \frac{x \cdot \sqrt{x^2 + 9}}{x^2 + 9} = 36</cmath>
 +
Simplifying, we get
 +
 +
<cmath>4x^4 - 48x^3 + 216x^2 - 432x + 324</cmath>
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 +
Now, we apply the quartic formula to get
 +
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<cmath>x = 6 \pm 3 \sqrt{3}</cmath>.
 +
 +
We can easily see that <math>x = 6 + 3 \sqrt{3}</math> is an invalid solution. Thus, <math>x = 6 - 3 \sqrt{3}</math>.
 +
 +
Finally, since <math>\tan (\theta) = \frac{3}{6 - 3 \sqrt{3}} = 2 + \sqrt{3}</math>, <math>\theta = \frac{5 + 12n}{12} \pi</math>, where <math>n</math> is any integer. Converting to degrees, we have that <math>\theta = 75 + 180n</math>. Since <math>0 < \theta < 90</math>, we have that <math>\theta = \boxed{75}</math>. <math>\square</math>
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~ilovepi3.14
  
 
== See Also==
 
== See Also==

Revision as of 19:41, 24 November 2018

Problem

Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?

$\textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

Solution 1

[asy] unitsize(10mm); defaultpen(linewidth(.5pt)+fontsize(10pt)); dotfactor=3; pair A=(0,3), B=(6,3), C=(6,0), D=(0,0); pair M=(0.80385,3); draw(A--B--C--D--cycle); draw(M--C); draw(M--D); draw(anglemark(A,M,D)); draw(anglemark(D,M,C)); draw(anglemark(C,D,M)); pair[] ps={A,B,C,D,M}; dot(ps); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$M$",M,N); label("$6$",midpoint(C--M),SW); label("$6$",midpoint(A--B),N); label("$3$",midpoint(B--C),E); [/asy]

It is given that $\angle AMD \sim \angle CMD$. Since $\angle AMD$ and $\angle CDM$ are alternate interior angles and $\overline{AB} \parallel \overline{DC}$, $\angle AMD \cong \angle CDM \longrightarrow \angle CMD \cong \angle CDM$. Use the Base Angle Theorem to show $\overline{DC} \cong \overline{MC}$. We know that $ABCD$ is a rectangle, so it follows that $\overline{MC} = 6$. We notice that $\triangle BMC$ is a $30-60-90$ triangle, and $\angle BMC = 30^{\circ}$. If we let $x$ be the measure of $\angle AMD,$ then \begin{align*} 2x + 30 &= 180\\ 2x &= 150\\ x &= \boxed{\textbf{(E)} 75} \end{align*}

Solution 2

Let $\angle{DMC} = \angle{AMD} = \theta$. If we let $AM = x$, we have that $MD = \sqrt{x^2 + 9}$, by the Pythagorean Theorem, and similarily, $MC = \sqrt{x^2 - 12x + 45}$. Applying LOC, we see that \[2x^2 + 54 - 12x - 2 \sqrt{x^4 - 12x^3 + 54x^2 - 108x + 405} \cdot cos (\theta) = 36\] and \[tan (\theta) = \frac{3}{x}\]. YAY!!! We have two equations for two variables... that are terribly ugly. Well, we'll try to solve it. First of all, note that $sin (\theta) = \frac{3}{\sqrt{x^2+9}}$, so solving for $cos (\theta)$ in terms of $x$, we get that $cos (\theta) = \frac{x \cdot \sqrt{x^2 + 9}}{x^2 + 9}$. The equation now becomes

\[2x^2 + 54 - 12x - 2 \sqrt{x^4 - 12x^3 + 54x^2 - 108x + 405} \cdot \frac{x \cdot \sqrt{x^2 + 9}}{x^2 + 9} = 36\] Simplifying, we get

\[4x^4 - 48x^3 + 216x^2 - 432x + 324\]

Now, we apply the quartic formula to get

\[x = 6 \pm 3 \sqrt{3}\].

We can easily see that $x = 6 + 3 \sqrt{3}$ is an invalid solution. Thus, $x = 6 - 3 \sqrt{3}$.

Finally, since $\tan (\theta) = \frac{3}{6 - 3 \sqrt{3}} = 2 + \sqrt{3}$, $\theta = \frac{5 + 12n}{12} \pi$, where $n$ is any integer. Converting to degrees, we have that $\theta = 75 + 180n$. Since $0 < \theta < 90$, we have that $\theta = \boxed{75}$. $\square$

~ilovepi3.14

See Also

2011 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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All AMC 10 Problems and Solutions

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