Chapter 12.4, Problem 4P

Figure 12.49 shows a trapezoid. This problem will help you find a formula for the area of the trapezoid and explain in several different ways why this formula is valid.

 

   Figure 12.49 A trapezoid.

1. Show how to combine two copies of the trapezoid in Figure 12.49 to make a parallelogram. Then use the formula for the area of a parallelogram to deduce a formula for the area of the trapezoid. Explain your reasoning.
2. Show how to cut off portions of the top part of the trapezoid and combine these portions with the bottom part of the trapezoid so as to make one or several parallelograms or rectangles, each of which has one side of length §h.Use this method to deduce a formula for the area of the trapezoid. Explain your reasoning.
3. By subdividing the trapezoid into two triangles, as shown in Figure 12.50 , find a formula in terms of a, b, and h for the area of the trapezoid, and explain why your formula is valid.

 

   Figure 12.50 A subdivided trapezoid.

Besides the methods of parts (a), (b), and (c) of this problem, there are still other methods for deducing the formula for the area of a trapezoid by using the moving and additivity principles. Find another method. Explain your reasoning