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

m (moved diagram above answer choices)
 
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==Problem==
+
== Problem ==
 
 
 
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>?
 
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>?
 
<math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad </math>
 
  
 
<asy>
 
<asy>
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D=(50,0);
 
D=(50,0);
 
E = (40,50);
 
E = (40,50);
  draw(A--B);
+
draw(A--B);
  draw(B--E);
+
draw(B--E);
  draw(E--C);
+
draw(E--C);
 
draw(C--D);
 
draw(C--D);
 
draw(D--A);
 
draw(D--A);
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label("D",D,SE);
 
label("D",D,SE);
 
label("E",E,N);
 
label("E",E,N);
 
 
</asy>
 
</asy>
  
==Solution==
+
<math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad </math>
  
 +
== Solution ==
 
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>
  
==Video Solution==
+
== Video Solution ==
 
https://www.youtube.com/watch?v=2vf843cvVzo?t=0
 
https://www.youtube.com/watch?v=2vf843cvVzo?t=0
 
~sugar_rush
 
~sugar_rush

Latest revision as of 13:14, 19 January 2021

Problem

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

[asy] pair A,B,C,D,E; A=(0,0); B=(0,50); C=(50,50); D=(50,0); E = (40,50); 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]

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

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 $E$

Video Solution

https://www.youtube.com/watch?v=2vf843cvVzo?t=0 ~sugar_rush

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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