Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2010 AMC 12A Problems/Problem 3"

m (Semi-automated contest formatting - script by azjps)
m (apologies for that)
Line 1: Line 1:
−
SCRIPT PREVIEW BY AZJPS
 
−
WILL AUTO-SUBMIT UNLESS YOU MOVE FROM WINDOW
 
 
== Problem ==
 
== Problem ==
 
Rectangle <math>ABCD</math>, pictured below, shares <math>50\%</math> of its area with square <math>EFGH</math>. Square <math>EFGH</math> shares <math>20\%</math> of its area with rectangle <math>ABCD</math>. What is <math>\frac{AB}{AD}</math>?
 
Rectangle <math>ABCD</math>, pictured below, shares <math>50\%</math> of its area with square <math>EFGH</math>. Square <math>EFGH</math> shares <math>20\%</math> of its area with rectangle <math>ABCD</math>. What is <math>\frac{AB}{AD}</math>?
Line 31: Line 29:
 
Let <math>EF = FG = GF = HE = s</math>, let <math>AD = BC = h</math>, and let <math>AB = CD = x</math>.
 
Let <math>EF = FG = GF = HE = s</math>, let <math>AD = BC = h</math>, and let <math>AB = CD = x</math>.
  
−
<cmath>\begin{align*}&\begin{align*}&\begin{align*}&\begin{align*}&0.2 \cdot s^2 = hs\\
+
<cmath>\begin{align*}0.2 \cdot s^2 = hs\\
 
  &s = 5h\\
 
  &s = 5h\\
 
  &0.5 \cdot hx = hs\\
 
  &0.5 \cdot hx = hs\\
 
  &x = 2s = 10h\\
 
  &x = 2s = 10h\\
−
  &\frac{AB}{AD} = \frac{x}{h} = \boxed{10\ \textbf{(E)}}\end{align*}\end{align*}\end{align*}\end{align*}</cmath>
+
  &\frac{AB}{AD} = \frac{x}{h} = \boxed{10\ \textbf{(E)}}\end{align*}</cmath>
  
 
=== Solution 2 ===
 
=== Solution 2 ===

Revision as of 21:53, 25 February 2010

Problem

Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?

[asy] unitsize(1mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(0,25)--(25,25)--(25,0)--cycle); fill((0,20)--(0,15)--(25,15)--(25,20)--cycle,gray); draw((0,15)--(0,20)--(25,20)--(25,15)--cycle); draw((25,15)--(25,20)--(50,20)--(50,15)--cycle); label("$A$",(0,20),W); label("$B$",(50,20),E); label("$C$",(50,15),E); label("$D$",(0,15),W); label("$E$",(0,25),NW); label("$F$",(25,25),NE); label("$G$",(25,0),SE); label("$H$",(0,0),SW); [/asy]

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

Solution

Solution 1

Let $EF = FG = GF = HE = s$, let $AD = BC = h$, and let $AB = CD = x$.

\begin{align*}0.2 \cdot s^2 = hs\\ &s = 5h\\ &0.5 \cdot hx = hs\\ &x = 2s = 10h\\ &\frac{AB}{AD} = \frac{x}{h} = \boxed{10\ \textbf{(E)}}\end{align*}

Solution 2

The answer does not change if we shift $A$ to coincide with $E$, and add new horizontal lines to divide $EFGH$ into five equal parts:

[asy] unitsize(1mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((0,0)--(0,25)--(25,25)--(25,0)--cycle); fill((0,25)--(0,20)--(25,20)--(25,25)--cycle,gray); draw((25,20)--(25,25)--(50,25)--(50,20)--cycle); draw((0,5)--(25,5)); draw((0,10)--(25,10)); draw((0,15)--(25,15)); label("$A=E$",(0,25),W); label("$B$",(50,25),E); label("$C$",(50,20),E); label("$D$",(0,20),W); label("$F$",(25,25),NE); label("$G$",(25,0),SE); label("$H$",(0,0),SW); [/asy]

This helps us to see that $AD=a/5$ and $AB=2a$, where $a=EF$. Hence $\dfrac{AB}{AD}=\dfrac{2a}{a/5}=10$.

See also

2010 AMC 12A (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 12 Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12A_Problems/Problem_3&oldid=33654"