1. **Problem:** Verify that the diagonals of quadrilateral ABCD with vertices A(0,0), B(2,3), C(5,1), and D(3,-2) are perpendicular. 2. **Formula and rules:** - The diagonals are the line segments AC and BD. - To check if two lines are perpendicular, their slopes $m_1$ and $m_2$ must satisfy $m_1 \times m_2 = -1$. - Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. 3. **Calculate slope of diagonal AC:** $$m_{AC} = \frac{1 - 0}{5 - 0} = \frac{1}{5}$$ 4. **Calculate slope of diagonal BD:** $$m_{BD} = \frac{-2 - 3}{3 - 2} = \frac{-5}{1} = -5$$ 5. **Check product of slopes:** $$m_{AC} \times m_{BD} = \frac{1}{5} \times (-5) = -1$$ 6. Since the product of slopes is $-1$, the diagonals AC and BD are perpendicular. **Final answer:** The diagonals of quadrilateral ABCD are perpendicular to each other.