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Difference between revisions of "2009 AMC 12B Problems/Problem 16"

(New page: == Problem == Trapezoid <math>ABCD</math> has <math>AD||BC</math>, <math>BD = 1</math>, <math>\angle DBA = 23^{\circ}</math>, and <math>\angle BDC = 46^{\circ}</math>. The ratio <math>BC:...)
 
m (Solution 2 has nothing underneath it)
 
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\mathrm{(E)}\ \frac {14}{15}</math>
 
\mathrm{(E)}\ \frac {14}{15}</math>
  
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== Solution ==
+
== Solutions ==
  
 
=== Solution 1 ===
 
=== Solution 1 ===
−
Extend <math>\overline AB</math> and <math>\overline DC</math> to meet at <math>E</math>.  Then
+
Extend <math>\overline {AB}</math> and <math>\overline {DC}</math> to meet at <math>E</math>.  Then
  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
Line 19: Line 19:
 
\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>
−
 
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=== Solution 2 ===
 
−
Let <math>E</math> be the intersection of <math>\overline {BC}</math> and the line parallel to <math>\overline {AB}.</math>  By constuction <math>BE = AD</math> and <math>\angle BDE = 23^{\circ}</math>; it follows that <math>DE</math> is the bisector of the angle <math>BDC</math>. So by the Angle Bisector Theorem we get
 
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<cmath>CD = \frac {CD}{BD} = \frac {EC}{BE} = \frac {BC - BE}{BE} = \frac {BC}{AD} -1 = \frac 95 - 1 = \boxed{\frac 45}.</cmath>
 
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The answer is <math>\mathrm{(B)}</math>.
 
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2009|ab=B|num-b=15|num-a=17}}
 
{{AMC12 box|year=2009|ab=B|num-b=15|num-a=17}}
 +
{{MAA Notice}}

Latest revision as of 20:36, 16 September 2024

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 $CD$?


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

Solutions

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 $CD = \boxed{\frac 45}.$

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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