1. **Problem Statement:** We have a parallelogram LMNP with vertices L(-4,1), M(2,-4), N(3,-2), and P(-3,-1). We want to determine which additional information would prove LMNP is a rectangle. 2. **Key Concept:** A parallelogram is a rectangle if and only if one of its angles is a right angle (90 degrees). This happens when two adjacent sides are perpendicular. 3. **Check slopes of sides:** - Slope of LP: $$\frac{y_P - y_L}{x_P - x_L} = \frac{-1 - 1}{-3 - (-4)} = \frac{-2}{1} = -2$$ - Slope of PN: $$\frac{y_N - y_P}{x_N - x_P} = \frac{-2 - (-1)}{3 - (-3)} = \frac{-1}{6} = -\frac{1}{6}$$ 4. **Check perpendicularity:** Two lines are perpendicular if the product of their slopes is $$-1$$. - Product of slopes LP and PN: $$-2 \times -\frac{1}{6} = \frac{2}{6} = \frac{1}{3} \neq -1$$ 5. **Check other pairs:** - Slope of MN: $$\frac{-2 - (-4)}{3 - 2} = \frac{2}{1} = 2$$ - Slope of LM: $$\frac{-4 - 1}{2 - (-4)} = \frac{-5}{6}$$ - Product of slopes LM and MN: $$-\frac{5}{6} \times 2 = -\frac{10}{6} = -\frac{5}{3} \neq -1$$ 6. **Conclusion:** The only option that guarantees a right angle is when two sides are perpendicular, i.e., when $$LP \perp PN$$. **Final answer:** The information "LP ⊥ PN" proves LMNP is a rectangle.