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

Latest revision as of 12:10, 1 July 2023

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 is:

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

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

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

and solving for $b$ ,

$b = 8$ , which is $\boxed{\textbf{(E)}}$

Video Solution (CREATIVE THINKING)

https://youtu.be/wW_GidbHnnM

~Education, the Study of Everything

Video Solution

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

2013 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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

2013 AMC 12A (Problems • Answer Key • Resources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 32: / Line 32: @@
 We are given that the area of <math>\triangle ABE</math> is <math>40</math>, and that <math>AB = 10</math>.
-The area of a triangle:
+The area of a triangle is:
 <math>A = \frac{bh}{2}</math>
@@ Line 40: / Line 40: @@
 <math>40 = \frac{10b}{2}</math>
-and solving for b,
+and solving for <math>b</math>,
+<math>b = 8</math>, which is <math>\boxed{\textbf{(E)}}</math>
+==Video Solution (CREATIVE THINKING)==
+https://youtu.be/wW_GidbHnnM
+~Education, the Study of Everything
-<math>b = 8</math>, which is <math>E</math>
 == Video Solution ==
@@ Line 49: / Line 55: @@
 == See also ==
-{{AMC10 box|year=2013|ab=A|before=Problem 2|num-a=4}}
+{{AMC10 box|year=2013|ab=A|num-b=2|num-a=4}}
 {{AMC12 box|year=2013|ab=A|before=First Question|num-a=2}}

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