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Difference between revisions of "2014 AMC 10B Problems/Problem 21"

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==Solution==
 
==Solution==
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<asy>
 +
draw((0,0)--(33,0)--(23, -9.79795897)--(2, -9.79795897)--(0,0));
 +
label("A", (0,0), NW);
 +
label("B", (33,0), NE);
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label("C", (23,-9.79795897), SE);
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label("D", (2, -9.79795897), SW);
 +
draw((2, 0)--(2, -9.79795897));
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draw((23, 0)--(23,-9.79795897));
 +
label("E", (2,0), N);
 +
label("F", (23,0), N);
 +
</asy>
 +
 +
In the diagram, <math>\overline{DE} \perp \overline{AB}, \overline{FC} \perp \overline{AB}</math>.
 +
Denote <math>\overline{AE} = x</math> and <math>\overline{DE} = h</math>. In right triangle <math>AED</math>, we have from the Pythagorean theorem: <math>x^2+h^2=100</math>. Note that since <math>EF = DC</math>, we have <math>BF = 33-DC-x = 12-x</math>. Using the Pythagorean theorem in right triangle <math>BFC</math>, we have <math>(12-x)^2 + h^2 = 196</math>. We isolate the <math>h^2</math> term in both equations, getting <math>\begin{align*}h^2 &= 100-x^2\\h^2 &= 196-(12-x)^2\end{align}</math>. Setting these equal, we have <math>100-x^2 = 196 - 144 + 24x -x^2 \implies 24x = 48 \implies x = 2</math>. Now, we can determine that <math>h^2 = 100-4 \implies h = \sqrt{96}</math>. The two diagonals are <math>\overline{AC}</math> and <math>\overline{BD}</math>. Using the Pythagorean theorem again on <math>\bigtriangleup AFC</math> and <math>\bigtriangleup BED</math>, we can find these lengths to be <math>\sqrt{96+529} = 25</math> and <math>\sqrt{96+961} = \sqrt{1057}</math>. Obviously, <math>25</math> is the shorter length, and thus the answer is <math>\boxed{\textbf{(B) }25}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2014|ab=B|num-b=20|num-a=22}}
 
{{AMC10 box|year=2014|ab=B|num-b=20|num-a=22}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 17:46, 20 February 2014

Problem

Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline {CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$?

$\textbf{(A) }10\sqrt{6}\qquad\textbf{(B) }25\qquad\textbf{(C) }8\sqrt{10}\qquad\textbf{(D) }18\sqrt{2}\qquad\textbf{(E) }26$

Solution

[asy] draw((0,0)--(33,0)--(23, -9.79795897)--(2, -9.79795897)--(0,0)); label("A", (0,0), NW); label("B", (33,0), NE); label("C", (23,-9.79795897), SE); label("D", (2, -9.79795897), SW); draw((2, 0)--(2, -9.79795897)); draw((23, 0)--(23,-9.79795897)); label("E", (2,0), N); label("F", (23,0), N); [/asy]

In the diagram, $\overline{DE} \perp \overline{AB}, \overline{FC} \perp \overline{AB}$. Denote $\overline{AE} = x$ and $\overline{DE} = h$. In right triangle $AED$, we have from the Pythagorean theorem: $x^2+h^2=100$. Note that since $EF = DC$, we have $BF = 33-DC-x = 12-x$. Using the Pythagorean theorem in right triangle $BFC$, we have $(12-x)^2 + h^2 = 196$. We isolate the $h^2$ term in both equations, getting $\begin{align*}h^2 &= 100-x^2\\h^2 &= 196-(12-x)^2\end{align}$ (Error compiling LaTeX. Unknown error_msg). Setting these equal, we have $100-x^2 = 196 - 144 + 24x -x^2 \implies 24x = 48 \implies x = 2$. Now, we can determine that $h^2 = 100-4 \implies h = \sqrt{96}$. The two diagonals are $\overline{AC}$ and $\overline{BD}$. Using the Pythagorean theorem again on $\bigtriangleup AFC$ and $\bigtriangleup BED$, we can find these lengths to be $\sqrt{96+529} = 25$ and $\sqrt{96+961} = \sqrt{1057}$. Obviously, $25$ is the shorter length, and thus the answer is $\boxed{\textbf{(B) }25}$

See Also

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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 10 Problems and Solutions

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