(04.01 MC) Quadrilateral ABCD has coordinates \( A(3,-5), B(5,-2), C(10,-4), D(8,-7) \). Quadrilateral ABCD is a rectangle, because opposite sides are congruent and adjacent sides are perpendicular square, because all four sides are congruent and adjacent sides are perpendicular parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular rhombus, because all four sides are congruent and adjacent sides are not perpendicular

To determine the type of quadrilateral ABCD based on the given coordinates \( A(3,-5), B(5,-2), C(10,-4), D(8,-7) \), we will follow these steps: 1. **Calculate the lengths of the sides** to check for congruence. 2. **Calculate the slopes of the sides** to check for perpendicularity. ### Step 1: Calculate the lengths of the sides The length of a side between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Let's calculate the lengths of the sides \( AB, BC, CD, \) and \( DA \). - Length of \( AB \): \[ AB = \sqrt{(5 - 3)^2 + (-2 + 5)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \] - Length of \( BC \): \[ BC = \sqrt{(10 - 5)^2 + (-4 + 2)^2} = \sqrt{(5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \] - Length of \( CD \): \[ CD = \sqrt{(8 - 10)^2 + (-7 + 4)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \] - Length of \( DA \): \[ DA = \sqrt{(3 - 8)^2 + (-5 + 7)^2} = \sqrt{(-5)^2 + (2)^2} = \sqrt{25 + 4} = \sqrt{29} \] ### Step 2: Check for congruence From the calculations: - \( AB = \sqrt{13} \) - \( CD = \sqrt{13} \) - \( BC = \sqrt{29} \) - \( DA = \sqrt{29} \) We see that opposite sides are congruent: - \( AB = CD \) - \( BC = DA \) ### Step 3: Calculate the slopes of the sides The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's calculate the slopes of the sides \( AB, BC, CD, \) and \( DA \). - Slope of \( AB \): \[ m_{AB} = \frac{-2 + 5}{5 - 3} = \frac{3}{2} \] - Slope of \( BC \): \[ m_{BC} = \frac{-4 + 2}{10 - 5} = \frac{-2}{5} \] - Slope of \( CD \): \[ m_{CD} = \frac{-7 + 4}{8 - 10} = \frac{-3}{-2} = \frac{3}{2} \] - Slope of \( DA \): \[ m_{DA} = \frac{-5 + 7}{3 - 8} = \frac{2}{-5} = -\frac{2}{5} \] ### Step 4: Check for perpendicularity Two lines are perpendicular if the product of their slopes is \(-1\). - Check \( AB \) and \( BC \): \[ m_{AB} \cdot m_{BC} = \frac{3}{2} \cdot \left(-\frac{2}{5}\right) = -\frac{3}{5} \quad (\text{not perpendicular}) \] - Check \( BC \) and \( CD \): \[ m_{BC} \cdot m_{CD} = \left(-\frac{2}{5}\right) \cdot \frac{3}{2} = -\frac{3}{5} \quad (\text{not perpendicular}) \] - Check \( CD \) and \( DA \): \[ m_{CD} \cdot m_{DA} = \frac{3}{2} \cdot \left(-\frac{2}{5}\right) = -\frac{3}{5} \quad (\text{not perpendicular}) \] - Check \( DA \) and \( AB \): \[ m_{DA} \cdot m_{AB} = -\frac{2}{5} \cdot \frac{3}{2} = -\frac{3}{5} \quad (\text{not perpendicular}) \] ### Conclusion Since opposite sides are congruent but adjacent sides are not perpendicular, quadrilateral ABCD is a **parallelogram**. Thus, the answer is: **parallelogram, because opposite sides are congruent and adjacent sides are not perpendicular.**