Difference between revisions of "2008 AMC 8 Problems/Problem 4"
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| − | ==Problem== | + | == Problem == |
In the figure, the outer equilateral triangle has area <math>16</math>, the inner equilateral triangle has area <math>1</math>, and the three trapezoids are congruent. What is the area of one of the trapezoids? | In the figure, the outer equilateral triangle has area <math>16</math>, the inner equilateral triangle has area <math>1</math>, and the three trapezoids are congruent. What is the area of one of the trapezoids? | ||
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<asy> | <asy> | ||
size((70)); | size((70)); | ||
| Line 8: | Line 9: | ||
draw((9.4,9.7)--(5.6,3.25)); | draw((9.4,9.7)--(5.6,3.25)); | ||
</asy> | </asy> | ||
| + | |||
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7</math> | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7</math> | ||
| − | ==Solution== | + | == Solution == |
The area outside the small triangle but inside the large triangle is <math>16-1=15</math>. This is equally distributed between the three trapezoids. Each trapezoid has an area of <math>15/3 = \boxed{\textbf{(C)}\ 5}</math>. | The area outside the small triangle but inside the large triangle is <math>16-1=15</math>. This is equally distributed between the three trapezoids. Each trapezoid has an area of <math>15/3 = \boxed{\textbf{(C)}\ 5}</math>. | ||
| − | ==See Also== | + | == See Also == |
{{AMC8 box|year=2008|num-b=3|num-a=5}} | {{AMC8 box|year=2008|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 12:36, 19 October 2020
Problem
In the figure, the outer equilateral triangle has area 
, the inner equilateral triangle has area 
, and the three trapezoids are congruent. What is the area of one of the trapezoids?
![[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy]](http://latex.artofproblemsolving.com/e/3/4/e34c3dede3f2bef33570072537b95ad87b3894f5.png)
Solution
The area outside the small triangle but inside the large triangle is 
. This is equally distributed between the three trapezoids. Each trapezoid has an area of 
.
See Also
| 2008 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
