Difference between revisions of "1997 AHSME Problems/Problem 5"
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| + | ==Problem== | ||
| + | |||
| + | A rectangle with perimeter <math>176</math> is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? | ||
| + | <asy> | ||
| + | defaultpen(linewidth(.8pt)); | ||
| + | draw(origin--(0,3)--(4,3)--(4,0)--cycle); | ||
| + | draw((0,1)--(4,1)); | ||
| + | draw((2,0)--midpoint((0,1)--(4,1))); | ||
| + | real r = 4/3; | ||
| + | draw((r,3)--foot((r,3),(0,1),(4,1))); | ||
| + | draw((2r,3)--foot((2r,3),(0,1),(4,1)));</asy> | ||
| + | |||
| + | <math> \mathrm{(A)\ } 35.2 \qquad \mathrm{(B) \ }76 \qquad \mathrm{(C) \ } 80 \qquad \mathrm{(D) \ } 84 \qquad \mathrm{(E) \ }86 </math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | Let <math>l</math> represent the length of one of the smaller rectangles, and let <math>w</math> represent the width of one of the smaller rectangles, with <math>w < l</math>. | ||
| + | |||
| + | From the large rectangle, we see that the top has length <math>3w</math>, the right has length <math>l + w</math>, the bottom has length <math>2l</math>, and the left has length <math>l + 2</math>. | ||
| + | |||
| + | Since the perimeter of the large rectangle is <math>176</math>, we know that <math>172 = 3w + l + w + 2l + l + w</math>, or <math>172 = 5w + 4l</math> | ||
| + | |||
| + | From the top and bottom of the large rectangle, we know that <math>3w = 2l</math>, or <math>l = 1.5w</math>. | ||
| + | |||
| + | Plugging that into the first equation, we get <math>176 = 5w + 4(1.5)w</math> | ||
| + | |||
| + | <math>176 = 11w</math> | ||
| + | |||
| + | <math>w = 16</math> | ||
| + | |||
| + | <math>l = 1.5w = 24</math> | ||
| + | |||
| + | <math>P = 2l + 2w = 2(16 + 24) = 80</math>, and the answer is <math>\boxed{C}</math> | ||
| + | |||
== See also == | == See also == | ||
{{AHSME box|year=1997|num-b=4|num-a=6}} | {{AHSME box|year=1997|num-b=4|num-a=6}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 13:12, 5 July 2013
Problem
A rectangle with perimeter 
is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
![[asy] defaultpen(linewidth(.8pt)); draw(origin--(0,3)--(4,3)--(4,0)--cycle); draw((0,1)--(4,1)); draw((2,0)--midpoint((0,1)--(4,1))); real r = 4/3; draw((r,3)--foot((r,3),(0,1),(4,1))); draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]](http://latex.artofproblemsolving.com/3/3/7/337a1fffee9d91efb37d484826230d83c48d36e3.png)
Solution
Let 
represent the length of one of the smaller rectangles, and let 
represent the width of one of the smaller rectangles, with 
.
From the large rectangle, we see that the top has length 
, the right has length 
, the bottom has length 
, and the left has length 
.
Since the perimeter of the large rectangle is , we know that 
, or 
From the top and bottom of the large rectangle, we know that 
, or 
.
Plugging that into the first equation, we get 
, and the answer is 
See also
| 1997 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
