2016 AMC 12B Problems/Problem 10

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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

Note that the slope of $PQ$ is $\frac{a-b}{b-a}=-1$ and the slope of $PS$ is $\frac{b+a}{a+b}=1$. Hence, $PQ\perp PS$ and we can similarly prove that the other angles are right angles. This means that $PQRS$ is a rectangle. 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 $a + b = \boxed{\textbf{(A)}\ 4}$.

Video Solution by TheBeautyofMath

https://www.youtube.com/watch?v=Eq2A2TTahqU with a second example of Shoelace Theorem done after this problem

~IceMatrix

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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