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The points $(1,3)$ and $(5,1)$ are two opposite vertices of a rectangle. The other two vertices lie on the line $\mathrm{y}=2 \mathrm{x}+\mathrm{c}$. What is the value of c?
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The correct answer is:
$-4$
Given, opposite vertices of rectangle are $\mathrm{A}(1,3)$ and $\mathrm{C}$ $(5,1)$
We know, diagonals of rectangle bisect each other. So, midpoint of AC lies on line $\mathrm{y}=2 \mathrm{x}+\mathrm{c}$.
Mid point of $\mathrm{AC}=\left(\frac{1+5}{2}, \frac{3+1}{2}\right)=\left(\frac{6}{2}, \frac{4}{2}\right)=(3,2)$
$\mathrm{y}=2 \mathrm{x}+\mathrm{c} \Rightarrow 2=2(3)+\mathrm{c}$
$\Rightarrow c=2-6=-4$
We know, diagonals of rectangle bisect each other. So, midpoint of AC lies on line $\mathrm{y}=2 \mathrm{x}+\mathrm{c}$.
Mid point of $\mathrm{AC}=\left(\frac{1+5}{2}, \frac{3+1}{2}\right)=\left(\frac{6}{2}, \frac{4}{2}\right)=(3,2)$
$\mathrm{y}=2 \mathrm{x}+\mathrm{c} \Rightarrow 2=2(3)+\mathrm{c}$
$\Rightarrow c=2-6=-4$
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