Difference between revisions of "2016 AMC 10A Problems/Problem 11"
Math101010 (talk | contribs) (Created page with "What is the area of the shaded region of the given <math>8 \times 5</math> rectangle? <asy> size(6cm); defaultpen(fontsize(9pt)); draw((0,0)--(8,0)--(8,5)--(0,5)--cycle); fi...") |
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<math>\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8</math> | <math>\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8</math> | ||
+ | |||
+ | First, split the rectangle into <math>4</math> triangles: | ||
+ | <asy> | ||
+ | |||
+ | size(6cm); | ||
+ | defaultpen(fontsize(9pt)); | ||
+ | draw((0,0)--(8,0)--(8,5)--(0,5)--cycle); | ||
+ | filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8)); | ||
+ | |||
+ | label("$1$",(1/2,5),dir(90)); | ||
+ | label("$7$",(9/2,5),dir(90)); | ||
+ | |||
+ | label("$1$",(8,1/2),dir(0)); | ||
+ | label("$4$",(8,3),dir(0)); | ||
+ | |||
+ | label("$1$",(15/2,0),dir(270)); | ||
+ | label("$7$",(7/2,0),dir(270)); | ||
+ | |||
+ | label("$1$",(0,9/2),dir(180)); | ||
+ | label("$4$",(0,2),dir(180)); | ||
+ | |||
+ | draw((0,5)--(8,0)); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | The bases of these triangles are all <math>1</math>, and their heights are <math>4</math>, <math>\frac{5}{2}</math>, <math>4</math>, and <math>\frac{5}{2}</math>. Thus, their areas are <math>2</math>, <math>\frac{5}{4}</math>, <math>2</math>, and <math>\frac{5}{4}</math>, which add to the area of the shaded region, which is <math>\boxed{6\frac{1}{2}}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2016|ab=A|num-b=10|num-a=12}} | ||
+ | {{MAA Notice}} |
Revision as of 22:03, 3 February 2016
What is the area of the shaded region of the given rectangle?
First, split the rectangle into triangles:
The bases of these triangles are all , and their heights are , , , and . Thus, their areas are , , , and , which add to the area of the shaded region, which is .
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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 |
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