2. Dirk draws quadrilateral \( R S T U \) with vertices \( R(-1,2), S(4,2), T(5,-1) \), and \( U(-2,-1) \). Which is the best way to classify the quadrilateral?

To classify the quadrilateral \( RSTU \), we can analyze the lengths of its sides and the slopes of its sides to determine its properties. 1. **Calculate the lengths of the sides:** - The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] We will calculate the lengths of each side: - Length of \( RS \): \[ d_{RS} = \sqrt{(4 - (-1))^2 + (2 - 2)^2} = \sqrt{(4 + 1)^2 + 0^2} = \sqrt{5^2} = 5 \] - Length of \( ST \): \[ d_{ST} = \sqrt{(5 - 4)^2 + (-1 - 2)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \] - Length of \( TU \): \[ d_{TU} = \sqrt{(-2 - 5)^2 + (-1 - (-1))^2} = \sqrt{(-7)^2 + 0^2} = \sqrt{49} = 7 \] - Length of \( UR \): \[ d_{UR} = \sqrt{(-1 - (-2))^2 + (2 - (-1))^2} = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \] 2. **Summarize the lengths:** - \( RS = 5 \) - \( ST = \sqrt{10} \) - \( TU = 7 \) - \( UR = \sqrt{10} \) 3. **Check for parallel sides:** - Calculate the slopes of each side: - Slope of \( RS \): \[ m_{RS} = \frac{2 - 2}{4 - (-1)} = \frac{0}{5} = 0 \] - Slope of \( ST \): \[ m_{ST} = \frac{-1 - 2}{5 - 4} = \frac{-3}{1} = -3 \] - Slope of \( TU \): \[ m_{TU} = \frac{-1 - (-1)}{-2 - 5} = \frac{0}{-7} = 0 \] - Slope of \( UR \): \[ m_{UR} = \frac{2 - (-1)}{-1 - (-2)} = \frac{3}{1} = 3 \] 4. **Analyze the slopes:** - \( RS \) and \( TU \) are horizontal (slope = 0), indicating they are parallel. - \( ST \) and \( UR \) have different slopes, indicating they are not parallel. 5. **Conclusion:** Since \( RS \) is parallel to \( TU \) and the lengths of \( ST \) and \( UR \) are equal, quadrilateral \( RSTU \) is a trapezoid. Thus, the best way to classify the quadrilateral \( RSTU \) is as a **trapezoid**.