Question:

Step 1: Identify the key property of rectangles. In a rectangle, the diagonals bisect each other. This means the center of the rectangle is located at the midpoint of any diagonal. Since $(2, 5)$ and $(5, 1)$ are opposite vertices, they form a diagonal of the rectangle. Step 2: Calculate the midpoint of the diagonal. The midpoint of the diagonal joining $(2, 5)$ and $(5, 1)$ is: $$\text{Midpoint} = \left(\frac{2+5}{2}, \frac{5+1}{2}\right) = \left(\frac{7}{2}, 3\right)$$ This point is the center of the rectangle. Step 3: Understand the constraint on the other two vertices. The other two vertices of the rectangle also form a diagonal, and this diagonal must also pass through the center of the rectangle at $\left(\frac{7}{2}, 3\right)$. Since these two vertices lie on the line $y = 2x + k$, the center of the rectangle must also lie on this line. Step 4: Substitute the center coordinates into the line equation. Since the center $\left(\frac{7}{2}, 3\right)$ lies on the line $y = 2x + k$, we substitute: $$3 = 2 \cdot \frac{7}{2} + k$$ $$3 = 7 + k$$ $$k = 3 - 7 = -4$$ Step 5: State the final answer. The value of $k$ is $\boxed{-4}$. The correct answer is **Option 3: -4**.