Question

The points \( (2, 5) \) and \( (5, 1) \) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line \( y = 2x + k \), then the value of \( k \) is:

• A. \( 4 \)
• B. \( 3 \)
• C. \( -4 \)
• D. \( -3 \)
• E. \( 1 \)

Hint:

For any parallelogram (including rectangles and squares), the center of the figure is the midpoint of the diagonals. If a line contains one diagonal, it must contain this center point.

Answer 1

The Correct Option is C

Concept: In a rectangle, the diagonals bisect each other. This means the midpoint of the diagonal joining the two given opposite vertices must also be the midpoint of the diagonal joining the other two vertices. Since the other two vertices lie on the line \( y = 2x + k \), their midpoint (which is the center of the rectangle) must also lie on that line.

Step 1: Find the midpoint of the given opposite vertices.
The vertices are \( A(2, 5) \) and \( C(5, 1) \). Midpoint \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \) \[ M = \left( \frac{2 + 5}{2}, \frac{5 + 1}{2} \right) = \left( \frac{7}{2}, 3 \right) \]

Step 2: Substitute the midpoint into the equation of the line.
The line \( y = 2x + k \) must pass through the midpoint \( M(3.5, 3) \): \[ 3 = 2\left( \frac{7}{2} \right) + k \] \[ 3 = 7 + k \]

Step 3: Solve for \( k \).
\[ k = 3 - 7 = -4 \]