Difference between revisions of "2005 AMC 8 Problems/Problem 13"

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== Problem ==
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The area of polygon <math> ABCDEF</math> is 52 with <math> AB=8</math>, <math>BC=9</math> and <math>FA=5</math>. What is <math> DE+EF</math>?
 
 
 
<asy>
 
pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4);
 
draw(a--b--c--d--e--f--cycle);
 
draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a);
 
draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b);
 
draw(shift(-.25,0)*c--shift(-.25,.25)*c--shift(0,.25)*c);
 
draw(shift(.25,0)*d--shift(.25,.25)*d--shift(0,.25)*d);
 
draw(shift(.25,0)*f--shift(.25,.25)*f--shift(0,.25)*f);
 
label("$A$", a, NW);
 
label("$B$", b, NE);
 
label("$C$", c, SE);
 
label("$D$", d, SW);
 
label("$E$", e, SW);
 
label("$F$", f, SW);
 
label("5", (0,6.5), W);
 
label("8", (4,9), N);
 
label("9", (8, 4.5), E);
 
</asy>
 
 
 
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
 
 
 
== Solution ==
 
Notice that <math>AF + DE = BC</math>, so <math>DE=4</math>. Let <math>O</math> be the intersection of the extensions of <math>AF</math> and <math>DC</math>, which makes rectangle <math>ABCO</math>. The area of the polygon is the area of <math>FEDO</math> subtracted from the area of <math>ABCO</math>.
 
 
 
<cmath>\text{Area} = 52 = 8 \cdot 9- EF \cdot 4</cmath>
 
 
 
Solving for the unknown, <math>EF=5</math>, therefore <math>DE+EF=4+5=\boxed{\textbf{(C)}\ 9}</math>.
 
 
 
== Solution 2 ==
 
Similar to solution 1, <math>AF + DE = BC</math>, so <math>DE=4</math>. Split the polygon into two rectangles by extending the <math>DE</math> so it intersects <math>AB</math>. Let's say the length of <math>FE</math> is equal to <math>x</math>. We can form the equation: <cmath>5x + 9(8-x) = 52</cmath>. We see that <math>x = 5</math>, so we can add <math>5 + 4 = \boxed{\textbf{(C)}\ 9}</math>.
 
- ArjunBhumula
 
 
 
==Video Solution by OmegaLearn==
 
https://youtu.be/abSgjn4Qs34?t=1908
 
 
 
~ pi_is_3.14
 
 
 
== See Also ==
 
{{AMC8 box|year=2005|num-b=12|num-a=14}}
 
{{MAA Notice}}
 

Revision as of 17:16, 5 May 2025

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