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

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{{duplicate|[[2009 AMC 12A Problems|2009 AMC 12A #20]] and [[2009 AMC 10A Problems|2009 AMC 10A #23]]}}
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__TOC__
 
== Problem ==
 
== Problem ==
 
Convex quadrilateral <math>ABCD</math> has <math>AB = 9</math> and <math>CD = 12</math>.  Diagonals <math>AC</math> and <math>BD</math> intersect at <math>E</math>, <math>AC = 14</math>, and <math>\triangle AED</math> and <math>\triangle BEC</math> have equal areas.  What is <math>AE</math>?
 
Convex quadrilateral <math>ABCD</math> has <math>AB = 9</math> and <math>CD = 12</math>.  Diagonals <math>AC</math> and <math>BD</math> intersect at <math>E</math>, <math>AC = 14</math>, and <math>\triangle AED</math> and <math>\triangle BEC</math> have equal areas.  What is <math>AE</math>?
  
 
<math>\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</math>
 
<math>\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</math>
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[[Category: Introductory Geometry Problems]]
  
__TOC__
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== Solution 1 ==
== Solution ==
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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}\times AC</math>, so <math>AE = \frac {3}{7}\times 14 = 6\ \boxed{\textbf{(E)}}</math>.
=== 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}AD = 6\ \boxed{\textbf{(E)}}</math>.
 
  
 
<center><asy>pathpen = linewidth(0.7);pointpen = black;
 
<center><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("14",(C+D)/2);
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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);
 
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></center>
 
</asy></center>
  
=== 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  
  
 
<center><cmath>(AE)(ED) = (BE)(EC)\quad \Longrightarrow\quad \frac {AE}{EC} = \frac {BE}{ED}.</cmath></center>
 
<center><cmath>(AE)(ED) = (BE)(EC)\quad \Longrightarrow\quad \frac {AE}{EC} = \frac {BE}{ED}.</cmath></center>
  
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>AE = 6</math>, <math>EC = 8\ \textbf{(E)}</math>.
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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>.
 +
 
 +
==Solution 3==
 +
The easiest way for the areas of the triangles to be equal would be if they were congruent [https://artofproblemsolving.com/wiki/index.php/Constraints_Strategy]. A way for that to work would be if <math>ABCD</math> were simply an isosceles trapezoid! Since <math>AC = 14</math> and <math>AE:EC = 3:4</math> (look at the side lengths and you'll know why!), <math>\boxed{AE = 6}</math>
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 +
==Solution 4 (Easiest Way)==
 +
Using the fact that <math>\triangle AED = \triangle BEC</math> and the fact that <math>\triangle AEB \sim \triangle EDC</math> (which should be trivial given the two equal triangles) we have that
 +
 
 +
<cmath>\frac{AE}{DC} = \frac{BE}{EC} = \frac{9}{12}</cmath>
 +
 
 +
We know that <math>DC=EC,</math> so we have
 +
 
 +
<cmath>\frac{AE}{EC} = \frac{BE}{EC} = \frac{3}{4}</cmath>
 +
 
 +
Thus
 +
 
 +
<cmath>\frac{AE}{EC} = \frac{3}{4}</cmath>
 +
 
 +
But <math>EC = 14 - AE</math> so we have
 +
 
 +
<cmath>\frac{AE}{14 - AE} = \frac{3}{4}</cmath>
 +
 
 +
Simplifying gives <math>AE = \boxed{6}.</math>
 +
 
 +
~mathboy282
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 +
===Note===
 +
 
 +
They are similar triangles because common sense. The two triangles that are equal in area implies that <math>AB</math> is parallel to <math>DC</math> which implies that <math>\angle{EAB} = \angle{CDE}</math> and <math>\angle{EBA} = \angle{DCE}.</math> Furthermore, since <math>\angle{AEB} = \angle{DEC}</math> (which is obvious why). By AAA similarity, <math>\triangle AEB \sim \triangle EDC.</math>
 +
 
 +
~mathboy282
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2009|ab=A|num-b=19|num-a=21}}
 
{{AMC12 box|year=2009|ab=A|num-b=19|num-a=21}}
 +
{{AMC10 box|year=2009|ab=A|num-b=22|num-a=24}}
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Revision as of 22:11, 2 November 2021

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 $3: 4$. $AE = \frac {3}{7}\times AC$, so $AE = \frac {3}{7}\times 14 = 6\ \boxed{\textbf{(E)}}$.

[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

The easiest way for the areas of the triangles to be equal would be if they were congruent [1]. A way for that to work would be if $ABCD$ were simply an isosceles trapezoid! Since $AC = 14$ and $AE:EC = 3:4$ (look at the side lengths and you'll know why!), $\boxed{AE = 6}$

Solution 4 (Easiest Way)

Using the fact that $\triangle AED = \triangle BEC$ and the fact that $\triangle AEB \sim \triangle EDC$ (which should be trivial given the two equal triangles) we have that

\[\frac{AE}{DC} = \frac{BE}{EC} = \frac{9}{12}\]

We know that $DC=EC,$ so we have

\[\frac{AE}{EC} = \frac{BE}{EC} = \frac{3}{4}\]

Thus

\[\frac{AE}{EC} = \frac{3}{4}\]

But $EC = 14 - AE$ so we have

\[\frac{AE}{14 - AE} = \frac{3}{4}\]

Simplifying gives $AE = \boxed{6}.$

~mathboy282

Note

They are similar triangles because common sense. The two triangles that are equal in area implies that $AB$ is parallel to $DC$ which implies that $\angle{EAB} = \angle{CDE}$ and $\angle{EBA} = \angle{DCE}.$ Furthermore, since $\angle{AEB} = \angle{DEC}$ (which is obvious why). By AAA similarity, $\triangle AEB \sim \triangle EDC.$

~mathboy282

See also

2009 AMC 12A (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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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