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The points $\left ( 2,5 \right )$ and $\left ( 6,3 \right )$ are two end points of a diagonal of a rectangle. If the other diagonal has the equation $y=3x+c$, then $c$ is

  1. $-5$
  2. $-6$
  3. $-7$
  4. $-8$
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We can draw the diagram for better understanding.



 Let the midpoint of the rectangle be $M(x,y)$.

In a rectangle, diagonals bisect each other. So, one diagonal should pass through the midpoint of the other.

Midpoint  $M(x,y)=M\left(\frac{2+6}{2},\frac{5+3}{2}\right) = M\left(\frac{8}{2},\frac{8}{2}\right) = M(4,4)$

The other diagonal,  $y=3x+c$  should also pass through $(4,4):$

Now, $4 = 3(4) + c$

$\Rightarrow  c = 4-12$

$\Rightarrow  \boxed{c = -8}$

$\therefore$ The value of $c$ is $ -8.$

Correct Answer $:\text{D}$

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