Find the exact length of the midsegment of the trapezoid with the vertices \( S(-2,4), T(-2,-4), U(3,-2), V(13,10) \). The length of the midsegment is

To find the length of the midsegment of the trapezoid with vertices \( S(-2,4) \), \( T(-2,-4) \), \( U(3,-2) \), and \( V(13,10) \), we first need to identify the bases of the trapezoid. The midsegment of a trapezoid is the line segment that connects the midpoints of the two bases. The length of the midsegment can be calculated using the formula: \[ \text{Length of midsegment} = \frac{b_1 + b_2}{2} \] where \( b_1 \) and \( b_2 \) are the lengths of the two bases. ### Step 1: Identify the bases We need to determine which pairs of vertices form the bases. The vertices \( S \) and \( T \) are both vertical (same x-coordinate), and the vertices \( U \) and \( V \) are not vertical. We will check if \( ST \) and \( UV \) are parallel. 1. **Base \( ST \)**: - The coordinates are \( S(-2, 4) \) and \( T(-2, -4) \). - The length of \( ST \) is calculated as: \[ \text{Length of } ST = |y_2 - y_1| = |-4 - 4| = 8 \] 2. **Base \( UV \)**: - The coordinates are \( U(3, -2) \) and \( V(13, 10) \). - The length of \( UV \) is calculated as: \[ \text{Length of } UV = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(13 - 3)^2 + (10 - (-2))^2} \] \[ = \sqrt{(10)^2 + (12)^2} = \sqrt{100 + 144} = \sqrt{244} = 2\sqrt{61} \] ### Step 2: Calculate the length of the midsegment Now that we have the lengths of the bases \( ST \) and \( UV \), we can find the length of the midsegment: \[ \text{Length of midsegment} = \frac{8 + 2\sqrt{61}}{2} = 4 + \sqrt{61} \] Thus, the exact length of the midsegment of the trapezoid is: \[ \boxed{4 + \sqrt{61}} \]