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

Revision as of 18:18, 2 September 2024

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

$[asy] unitsize(10mm); defaultpen(linewidth(.5pt)+fontsize(10pt)); dotfactor=3; pair A=(0,3), B=(6,3), C=(6,0), D=(0,0); pair M=(0.80385,3); draw(A--B--C--D--cycle); draw(M--C); draw(M--D); draw(anglemark(A,M,D)); draw(anglemark(D,M,C)); pair[] ps={A,B,C,D,M}; dot(ps); label("$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("$3$",midpoint(B--C),E); label("$6$",midpoint(C--D),S); [/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 $\begin{align*} 2x + 30 &= 180\\ 2x &= 150\\ x &= \boxed{\textbf{(E)} 75} \end{align*}$

Easier Way to Continue

After finding $MC = 6,$ we can continue using trigonometry as follows.

We know that $\angle{BMC} = 180-2x$ and so $\sin (180-2x) = \frac{3}{6} = \frac{1}{2}$

It is obvious that $\sin (30) = \frac{1}{2}$ and so $180-2x=30.$

Solving, we have $x = \boxed{75}$

~mathboy282

Solution 2 (with trig, not recommended)

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 the law of cosine, 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 relatively difficult to deal with. 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

Solution 3(Easier Trig)

We have $DC=CM=6$ . By the Pythagorean Theorem, $BM=\sqrt{6^2-3^2}=3\sqrt{3}$ , and thus $AM=6-3\sqrt{3}$ , We have $\tan(AMD)=\frac{6-3\sqrt{3}}{3}=2+\sqrt{3}$ , or $\angle AMD=\boxed{75}$ ~awsomek

Solution 4 (elimination)

Let $\angle AMD=\angle DMC=\theta$ . Thus, $\angle BMC=180-2\theta\implies\angle MCB=2\theta-90$ . Since all angles should be positive, $\theta>45^\circ$ , narrowing the options to D and E. Trying $60^\circ$ (option D), $\Delta AMD$ is a 30-60-90 triangle. $AD=3$ , so it follows that $AM=\sqrt3$ . Since $\angle BMC=180-2\theta$ , $\angle BMC=60^\circ$ , too. However, that would imply that $\Delta MBC$ is also a $30-60-90$ triangle, which would, in turn, imply that $MB=3\sqrt3$ , since $BC=3$ . We know that $AM+MB=AB$ and $AB=6,$ but we know that $AM=\sqrt3$ and $MB=3\sqrt3$ . $AM+MB$ is clearly not $6$ , so this is not possible. Thus, the answer must be $\boxed{\textbf{(E)}~75}$ . ~ Technodoggo

@@ Line 6: / Line 6: @@
 [[Category: Introductory Geometry Problems]]
-==Solution==
+==Solution 1==
 <center><asy>
@@ Line 21: / Line 21: @@
 draw(anglemark(A,M,D));
 draw(anglemark(D,M,C));
-draw(anglemark(C,D,M));
 pair[] ps={A,B,C,D,M};
@@ Line 31: / Line 31: @@
 label("$M$",M,N);
 label("$6$",midpoint(C--M),SW);
-label("$6$",midpoint(A--B),N);
 label("$3$",midpoint(B--C),E);
+label("$6$",midpoint(C--D),S);
 </asy>
@@ Line 43: / Line 43: @@
 x &= \boxed{\textbf{(E)} 75}
 \end{align*}</cmath>
+===Easier Way to Continue===
+After finding <math>MC = 6,</math> we can continue using trigonometry as follows.
+We know that <math>\angle{BMC} = 180-2x</math> and so <math>\sin (180-2x) = \frac{3}{6} = \frac{1}{2}</math>
+It is obvious that <math>\sin (30) = \frac{1}{2}</math> and so <math>180-2x=30.</math>
+Solving, we have <math>x = \boxed{75}</math>
+~mathboy282
+==Solution 2 (with trig, not recommended)==
+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 the law of cosine, 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 relatively difficult to deal with. 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>
+Now, we apply the quartic formula to get
+<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>
+~ilovepi3.14
+==Solution 3(Easier Trig)==
+We have <math>DC=CM=6</math>. By the Pythagorean Theorem, <math>BM=\sqrt{6^2-3^2}=3\sqrt{3}</math>, and thus <math>AM=6-3\sqrt{3}</math>, We have <math>\tan(AMD)=\frac{6-3\sqrt{3}}{3}=2+\sqrt{3}</math>, or <math>\angle AMD=\boxed{75}</math>
+~awsomek
+== Solution 4 (elimination) ==
+Let <math>\angle AMD=\angle DMC=\theta</math>. Thus, <math>\angle BMC=180-2\theta\implies\angle MCB=2\theta-90</math>. Since all angles should be positive, <math>\theta>45^\circ</math>, narrowing the options to D and E. Trying <math>60^\circ</math> (option D), <math>\Delta AMD</math> is a 30-60-90 triangle. <math>AD=3</math>, so it follows that <math>AM=\sqrt3</math>.
+Since <math>\angle BMC=180-2\theta</math>, <math>\angle BMC=60^\circ</math>, too. However, that would imply that <math>\Delta MBC</math> is also a <math>30-60-90</math> triangle, which would, in turn, imply that <math>MB=3\sqrt3</math>, since <math>BC=3</math>. We know that <math>AM+MB=AB</math> and <math>AB=6,</math> but we know that <math>AM=\sqrt3</math> and <math>MB=3\sqrt3</math>. <math>AM+MB</math> is clearly not <math>6</math>, so this is not possible.
+Thus, the answer must be <math>\boxed{\textbf{(E)}~75}</math>.
+~ Technodoggo
 == See Also==

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