Identify the polygon that has vertices \( J(12,-4), U(0,-4) \), \( S(4,3) \), and \( T(8,3) \), and then find the perimeter and area of the polygon.

We first plot the points: - \( J(12,-4) \) - \( U(0,-4) \) - \( S(4,3) \) - \( T(8,3) \) Notice that \( J \) and \( U \) lie on the line \( y=-4 \) (the bottom) and \( S \) and \( T \) lie on the line \( y=3 \) (the top). Arranging the points in order along the convex hull gives the quadrilateral with vertices \[ U(0,-4),\ J(12,-4),\ T(8,3),\ S(4,3). \] This is a trapezoid with bases \( \overline{UJ} \) and \( \overline{ST} \). **Step 1. Identify the base lengths.** - The bottom base from \( U(0,-4) \) to \( J(12,-4) \) has length \[ b_1 = |12-0| = 12. \] - The top base from \( S(4,3) \) to \( T(8,3) \) has length \[ b_2 = |8-4| = 4. \] **Step 2. Find the lengths of the legs.** Calculate the distance between \( U(0,-4) \) and \( S(4,3) \): \[ \text{Leg}_1 = \sqrt{(4-0)^2 + (3-(-4))^2} = \sqrt{4^2 + 7^2} = \sqrt{16+49} = \sqrt{65}. \] Similarly, calculate the distance between \( J(12,-4) \) and \( T(8,3) \): \[ \text{Leg}_2 = \sqrt{(8-12)^2 + (3-(-4))^2} = \sqrt{(-4)^2 + 7^2} = \sqrt{16+49} = \sqrt{65}. \] **Step 3. Compute the perimeter.** The perimeter \( P \) is the sum of all side lengths: \[ P = b_1 + b_2 + \text{Leg}_1 + \text{Leg}_2 = 12 + 4 + \sqrt{65} + \sqrt{65} = 16 + 2\sqrt{65}. \] **Step 4. Calculate the area.** The area \( A \) of a trapezoid is given by: \[ A = \frac{1}{2} \left(b_1 + b_2\right) \times h, \] where \( h \) is the height. Here, the height is the vertical distance between the bases: \[ h = 3 - (-4) = 7. \] Thus, the area is: \[ A = \frac{1}{2} (12 + 4) \times 7 = \frac{1}{2} (16) \times 7 = 8 \times 7 = 56. \] **Summary** - The polygon is a trapezoid. - Perimeter: \(16 + 2\sqrt{65}\) - Area: \(56\)