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Difference between revisions of "2009 AMC 12B Problems/Problem 16"
m (→‎Solution 1)
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\end{align*}</cmath>
 
\end{align*}</cmath>
  
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Thus <math>\triangle BDE</math> is isosceles with <math>DE = BD</math>.  Because <math>\overline {AD} \parallel \overline {BC}</math>, it follows that the triangles <math>BCD</math> and <math>ADE</math> are similar.  Therefore
+
Thus <math>\triangle BDE</math> is isosceles with <math>DE = BD</math>.  Because <math>\overline {AD} \parallel \overline {BC}</math>, it follows that the triangles <math>BCE</math> and <math>ADE</math> are similar.  Therefore
 
<cmath>\frac 95 = \frac {BC}{AD} = \frac {CD + DE}{DE} = \frac {CD}{BD} + 1 = CD + 1,</cmath>
 
<cmath>\frac 95 = \frac {BC}{AD} = \frac {CD + DE}{DE} = \frac {CD}{BD} + 1 = CD + 1,</cmath>
 
so <math>CD = \boxed{\frac 45}.</math>
 
so <math>CD = \boxed{\frac 45}.</math>

Revision as of 23:56, 8 February 2010

Problem

Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$?


$\mathrm{(A)}\ \frac 79\qquad \mathrm{(B)}\ \frac 45\qquad \mathrm{(C)}\ \frac {13}{15}\qquad \mathrm{(D)}\ \frac 89\qquad \mathrm{(E)}\ \frac {14}{15}$

Solution

Solution 1

Extend $\overline {AB}$ and $\overline {DC}$ to meet at $E$. Then

\begin{align*} \angle BED &= 180^{\circ} - \angle EDB - \angle DBE\\ &= 180^{\circ} - 134^{\circ} -23^{\circ} = 23^{\circ}. \end{align*}

Thus $\triangle BDE$ is isosceles with $DE = BD$. Because $\overline {AD} \parallel \overline {BC}$, it follows that the triangles $BCE$ and $ADE$ are similar. Therefore \[\frac 95 = \frac {BC}{AD} = \frac {CD + DE}{DE} = \frac {CD}{BD} + 1 = CD + 1,\] so $CD = \boxed{\frac 45}.$

Solution 2

Let $E$ be the intersection of $\overline {BC}$ and the line through $D$ parallel to $\overline {AB}.$ By constuction $BE = AD$ and $\angle BDE = 23^{\circ}$; it follows that $DE$ is the bisector of the angle $BDC$. So by the Angle Bisector Theorem we get \[CD = \frac {CD}{BD} = \frac {EC}{BE} = \frac {BC - BE}{BE} = \frac {BC}{AD} -1 = \frac 95 - 1 = \boxed{\frac 45}.\] The answer is $\mathrm{(B)}$.

See also

2009 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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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