Difference between revisions of "2013 AMC 12A Problems/Problem 1"

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==Problem==
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Square <math> ABCD </math> has side length <math> 10 </math>. Point <math> E </math> is on <math> \overline{BC} </math>, and the area of <math> \bigtriangleup ABE </math> is <math> 40 </math>. What is <math> BE </math>?
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<math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad </math>
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<asy>
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pair A,B,C,D,E;
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A=(0,0);
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B=(0,50);
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C=(50,50);
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D=(50,0);
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E = (30,50);
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  draw(A--B);
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  draw(B--E);
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  draw(E--C);
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draw(C--D);
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draw(D--A);
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draw(A--E);
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dot(A);
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dot(B);
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dot(C);
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dot(D);
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dot(E);
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label("A",A,SW);
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label("B",B,NW);
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label("C",C,NE);
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label("D",D,SE);
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label("E",E,N);
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</asy>
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==Solution==
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We are given that the area of <math>\triangle ABE</math> is <math>40</math>, and that <math>AB = 10</math>.
 
We are given that the area of <math>\triangle ABE</math> is <math>40</math>, and that <math>AB = 10</math>.
  
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<math>b = 8</math>, which is <math>E</math>
 
<math>b = 8</math>, which is <math>E</math>
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== See also ==
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{{AMC12 box|year=2013|ab=A|before=First Question|num-a=2}}

Revision as of 18:25, 22 February 2013

Problem

Square $ABCD$ has side length $10$. Point $E$ is on $\overline{BC}$, and the area of $\bigtriangleup ABE$ is $40$. What is $BE$?

$\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad$

[asy] pair A,B,C,D,E; A=(0,0); B=(0,50); C=(50,50); D=(50,0); E = (30,50);    draw(A--B);    draw(B--E);    draw(E--C); draw(C--D); draw(D--A); draw(A--E); dot(A); dot(B); dot(C); dot(D); dot(E); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,N);  [/asy]

Solution

We are given that the area of $\triangle ABE$ is $40$, and that $AB = 10$.

The area of a triangle:

$A = \frac{bh}{2}$

Using $AB$ as the height of $\triangle ABE$,

$40 = \frac{10b}{2}$

and solving for b,

$b = 8$, which is $E$

See also

2013 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
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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