1. **State the problem:** We are given the vertices of quadrilateral LMNP as L(-1, 7), M(4, 9), N(8, -1), and P(3, -3). We need to classify LMNP as a parallelogram, rectangle, rhombus, or square using the distance formula. 2. **Recall the distance formula:** The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Calculate the lengths of all sides:** - $LM = \sqrt{(4 - (-1))^2 + (9 - 7)^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}$ - $MN = \sqrt{(8 - 4)^2 + (-1 - 9)^2} = \sqrt{4^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116}$ - $NP = \sqrt{(3 - 8)^2 + (-3 - (-1))^2} = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}$ - $PL = \sqrt{(-1 - 3)^2 + (7 - (-3))^2} = \sqrt{(-4)^2 + 10^2} = \sqrt{16 + 100} = \sqrt{116}$ 4. **Check opposite sides:** - $LM = NP = \sqrt{29}$ - $MN = PL = \sqrt{116}$ Opposite sides are equal, so LMNP is at least a parallelogram. 5. **Check diagonals to determine if rectangle or rhombus:** - $LN = \sqrt{(8 - (-1))^2 + (-1 - 7)^2} = \sqrt{9^2 + (-8)^2} = \sqrt{81 + 64} = \sqrt{145}$ - $MP = \sqrt{(3 - 4)^2 + (-3 - 9)^2} = \sqrt{(-1)^2 + (-12)^2} = \sqrt{1 + 144} = \sqrt{145}$ Diagonals are equal, which is a property of rectangles and squares. 6. **Check if all sides are equal for rhombus or square:** - Sides are $\sqrt{29}$ and $\sqrt{116}$, not all equal, so not a rhombus or square. 7. **Conclusion:** LMNP has opposite sides equal and diagonals equal, so it is a rectangle. **Final answer:** LMNP is a rectangle.