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

$-5$

$-6$

$-7$

$-8$

✅ Correct Option: 4

In any rectangle, the diagonals always bisect each other. This means they cut each other exactly in half - they meet at their common midpoint.

Since both diagonals pass through the same point (their common midpoint), we can use this shared point to find our unknown value.

The diagonal connects points $(2, 5)$ and $(6, 3)$ .

Using the midpoint formula: $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$

Midpoint = $\left(\dfrac{2 + 6}{2}, \dfrac{5 + 3}{2}\right)$

$= \left(\dfrac{8}{2}, \dfrac{8}{2}\right)$

$= (4, 4)$

Since diagonals bisect each other, the other diagonal $y = 3x + c$ must also pass through the point $(4, 4)$ .

If a line passes through a point, that point must satisfy the line's equation.

Substituting the point $(4, 4)$ into $y = 3x + c$ :

$4 = 3(4) + c$

$4 = 12 + c$

$c = 4 - 12 = -8$

Therefore, $c = -8$

This approach works because we leveraged the fundamental property of rectangles rather than trying to find all four vertices - much faster and more elegant!

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CAT Past Year Questions (Topic-Wise)

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