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Problem 6

# If the points $$(1,3)$$ and $$(5,1)$$ are two opposite vertices of a rectangle and the other two vertices lie on the line $$y=2 x+c$$, then the value of $$c$$ is : (a) $$-2$$ (b) 3 $$\begin{array}{ll}\text { (c) }-4 & \text { (d) } 5\end{array}$$

Expert verified
The value of c in the line equation where the other two vertices of the rectangle lie is -4, hence answer (c) is correct.
See the step by step solution

## Step 1: Identify the Properties of a Rectangle

A key thing to remember is that in a rectangle, diagonals are of equal length. Here, the two given points can be considered as endpoints of a diagonal of the rectangle.

## Step 2: Find the Midpoint

Calculate the midpoint of the diagonal line formed by the points (1, 3) and (5, 1). This can be done using the midpoint formula which is [(x1 + x2)/2, (y1 + y2)/2]. Substituting the given points into this formula results in the midpoint (3, 2).

## Step 3: Substitute into the Line Equation

Since the other two vertices lie on the line y=2x+c, the midpoint should also lie on this line. Therefore, substitute the x and y coordinates of the midpoint (3, 2) into the equation, resulting in the equation 2 = 2*3 + c.

## Step 4: Solve for c

Solving this equation for c, we get: c = 2 - 6, therefore c = -4

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