Difference between revisions of "2016 AMC 12B Problems/Problem 10"

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Solution by I_Dont_Do_Math
 
Solution by I_Dont_Do_Math
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==Solution 2==
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Solution by e_power_pi_times_i
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By the Shoelace Theorem, the area of the quadrilateral is <math>2a^2 - 2b^2</math>, so <math>a^2 - b^2 = 8</math>. Since <math>a</math> and <math>b</math> are integers, <math>a = 3</math> and <math>b = 1</math>, so <math>a + b = \boxed{\textbf{(A)}\ 4}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=B|num-b=9|num-a=11}}
 
{{AMC12 box|year=2016|ab=B|num-b=9|num-a=11}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:51, 22 February 2016

Problem

A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?

$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12  \qquad\textbf{(E)}\ 13$

Solution

By distance formula we have $(a-b)^2+(b-a)^2*2*(a+b)^2 = 256$. SImplifying we get $(a-b)(a+b) = 8$. Thus $a+b$ and $a-b$ have to be a factor of 8. The only way for them to be factors of $8$ and remain integers is if $a+b = 4$ and $a-b = 2$. So the answer is $\boxed{\textbf{(A)}\ 4}$

Solution by I_Dont_Do_Math

Solution 2

Solution by e_power_pi_times_i


By the Shoelace Theorem, the area of the quadrilateral is $2a^2 - 2b^2$, so $a^2 - b^2 = 8$. Since $a$ and $b$ are integers, $a = 3$ and $b = 1$, so $a + b = \boxed{\textbf{(A)}\ 4}$.

See Also

2016 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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

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