Final answer:
The student's question requires graphing trapezoid BATS based on given coordinates and calculating its area using the trapezoid area formula. The length of the sides is obtained using the distance formula, and TM is the height. The information is then substituted into the area formula to obtain the result.
Step-by-step explanation:
The question requires us to graph the trapezoid BATS with the given vertices and find its area. We graph the points B (-4, -1), A (-2,5), T (4,8), and S (8,5) on the Cartesian plane and draw the trapezoid by connecting these points. Point M (6, 4) lies on TS and is the foot of the perpendicular from T, making TM the height of the trapezoid.
The area of a trapezoid is given by the formula A = \( \frac{1}{2} (b_1 + b_2) h \), where \(b_1\) and \(b_2\) are the lengths of the two parallel sides, and \(h\) is the height. To find the lengths of the parallel sides (BA and TS), we calculate their distances using the distance formula, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). The height is the length of segment TM.
After performing the calculations, we substitute the lengths of the sides and the height into the area formula to get the area of the trapezoid BATS. This is a classical geometry and algebra question that integrates knowledge of coordinate geometry and the application of formulas.