Difference between revisions of "2013 AMC 12A Problems/Problem 1"
m (moved diagram above answer choices) |
|||
| (9 intermediate revisions by 6 users not shown) | |||
| Line 1: | Line 1: | ||
| − | + | == 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>? | ||
| − | + | <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> | ||
| − | + | <math>\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad </math> | |
| − | 40= | + | == Solution == |
| + | We are given that the area of <math>\triangle ABE</math> is <math>40</math>, and that <math>AB = 10</math>. | ||
| − | and solving for | + | The area of a triangle: |
| + | |||
| + | <math>A = \frac{bh}{2}</math> | ||
| + | |||
| + | Using <math>AB</math> as the height of <math>\triangle ABE</math>, | ||
| + | |||
| + | <math>40 = \frac{10b}{2}</math> | ||
| + | |||
| + | and solving for b, | ||
| + | |||
| + | <math>b = 8</math>, which is <math>E</math> | ||
| + | |||
| + | == Video Solution == | ||
| + | https://www.youtube.com/watch?v=2vf843cvVzo?t=0 | ||
| + | ~sugar_rush | ||
| + | |||
| + | == See also == | ||
| + | {{AMC12 box|year=2013|ab=A|before=First Question|num-a=2}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
| + | [[Category:Area Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 14:14, 19 January 2021
Contents
Problem
Square 
has side length 
. Point 
is on 
, and the area of 
is 
. What is 
?
![[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]](http://latex.artofproblemsolving.com/5/d/e/5deb6040701b50303c4593a48dd3b09550139cd9.png)
Solution
We are given that the area of 
is , and that 
.
The area of a triangle:
Using 
as the height of ,
and solving for b,
, which is
Video Solution
https://www.youtube.com/watch?v=2vf843cvVzo?t=0 ~sugar_rush
See also
| 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. 
