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Difference between revisions of "2009 AMC 12A Problems/Problem 20"

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== Solution ==
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== Solution 1 ==
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=== Solution 1 ===
 
 
Let <math>[ABC]</math> denote the area of triangle <math>ABC</math>. <math>[AED] = [BEC]</math>, so <math>[ABD] = [AED] + [AEB] = [BEC] + [AEB] = [ABC]</math>. Since triangles <math>ABD</math> and <math>ABC</math> share a base, they also have the same height and thus <math>\overline{AB}||\overline{CD}</math> and <math>\triangle{AEB}\sim\triangle{CED}</math> with a ratio of <math>3: 4</math>. <math>AE = \frac {3}{7}AC = 6\ \boxed{\textbf{(E)}}</math>.
 
Let <math>[ABC]</math> denote the area of triangle <math>ABC</math>. <math>[AED] = [BEC]</math>, so <math>[ABD] = [AED] + [AEB] = [BEC] + [AEB] = [ABC]</math>. Since triangles <math>ABD</math> and <math>ABC</math> share a base, they also have the same height and thus <math>\overline{AB}||\overline{CD}</math> and <math>\triangle{AEB}\sim\triangle{CED}</math> with a ratio of <math>3: 4</math>. <math>AE = \frac {3}{7}AC = 6\ \boxed{\textbf{(E)}}</math>.
  
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=== Solution 2 ===
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== Solution 2 ==
 
Using the sine area formula on triangles <math>AED</math> and <math>BEC</math>, as <math>\angle AED = \angle BEC</math>, we see that  
 
Using the sine area formula on triangles <math>AED</math> and <math>BEC</math>, as <math>\angle AED = \angle BEC</math>, we see that  
  
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Since <math>\angle AEB = \angle DEC</math>, triangles <math>AEB</math> and <math>DEC</math> are similar. Their ratio is <math>\frac {AB}{CD} = \frac {3}{4}</math>.  Since <math>AE + EC = 14</math>, we must have <math>EC = 8</math>, so <math>AE = 6\ \textbf{(E)}</math>.
 
Since <math>\angle AEB = \angle DEC</math>, triangles <math>AEB</math> and <math>DEC</math> are similar. Their ratio is <math>\frac {AB}{CD} = \frac {3}{4}</math>.  Since <math>AE + EC = 14</math>, we must have <math>EC = 8</math>, so <math>AE = 6\ \textbf{(E)}</math>.
  
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===Solution 3 (which won't work when justification is required)===
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==Solution 3 (which won't work when justification is required)==
 
Consider an isosceles trapezoid with opposite bases of <math>9</math> and <math>12</math> and a diagonal of length <math>14</math>. This trapezoid exists and satisfies all the conditions in the problem (the areas are congruent by symmetry). So, we can simply use this easier case to solve this problem, because the problem implies that the answer is invariant for all quadrilaterals satisfying the conditions.
 
Consider an isosceles trapezoid with opposite bases of <math>9</math> and <math>12</math> and a diagonal of length <math>14</math>. This trapezoid exists and satisfies all the conditions in the problem (the areas are congruent by symmetry). So, we can simply use this easier case to solve this problem, because the problem implies that the answer is invariant for all quadrilaterals satisfying the conditions.
  

Revision as of 22:58, 24 January 2017

The following problem is from both the 2009 AMC 12A #20 and 2009 AMC 10A #23, so both problems redirect to this page.

Problem

Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?

$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad \textbf{(E)}\ 6$

Solution 1

Let $[ABC]$ denote the area of triangle $ABC$. $[AED] = [BEC]$, so $[ABD] = [AED] + [AEB] = [BEC] + [AEB] = [ABC]$. Since triangles $ABD$ and $ABC$ share a base, they also have the same height and thus $\overline{AB}||\overline{CD}$ and $\triangle{AEB}\sim\triangle{CED}$ with a ratio of $3: 4$. $AE = \frac {3}{7}AC = 6\ \boxed{\textbf{(E)}}$.

[asy]pathpen = linewidth(0.7);pointpen = black; pair D=MP("D",(0,0)),C=MP("C",(12,0)),A=MP("A",C+14*expi(145*pi/180),N),B=MP("B",A+(9,0),N),E=IP(A--C,B--D);MP("9",(A+B)/2,N);MP("12",(C+D)/2); fill(A--D--E--cycle,rgb(0.8,0.8,0.8));fill(B--C--E--cycle,rgb(0.8,0.8,0.8));D(A--B--C--D--cycle);D(A--C);D(B--D);D(E); [/asy]

Solution 2

Using the sine area formula on triangles $AED$ and $BEC$, as $\angle AED = \angle BEC$, we see that

\[(AE)(ED) = (BE)(EC)\quad \Longrightarrow\quad \frac {AE}{EC} = \frac {BE}{ED}.\]

Since $\angle AEB = \angle DEC$, triangles $AEB$ and $DEC$ are similar. Their ratio is $\frac {AB}{CD} = \frac {3}{4}$. Since $AE + EC = 14$, we must have $EC = 8$, so $AE = 6\ \textbf{(E)}$.

Solution 3 (which won't work when justification is required)

Consider an isosceles trapezoid with opposite bases of $9$ and $12$ and a diagonal of length $14$. This trapezoid exists and satisfies all the conditions in the problem (the areas are congruent by symmetry). So, we can simply use this easier case to solve this problem, because the problem implies that the answer is invariant for all quadrilaterals satisfying the conditions.

Then, by similar triangles, the ratio of $AE$ to $EC$ is $3:4$, so $AE=\boxed{\textbf{(E)}6}$.

See also

2009 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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
2009 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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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