Question

Find the area of the trapezoid by decomposing it into other shapes.

Answer 1

To find the area of a trapezoid, we can decompose it into other shapes and then find the area of each shape separately.

Let's consider a trapezoid with parallel sides of lengths a and b, and height h.

First, we can decompose the trapezoid into a rectangle and two right triangles.

The rectangle has length a and height h. Its area is given by A_rectangle = a * h.

The two right triangles have base lengths (b - a)/2 and (b - a)/2, and height h. Their areas are given by A_triangle1 = 0.5 * ((b - a)/2) * h and A_triangle2 = 0.5 * ((b - a)/2) * h.

Now, we can find the total area of the trapezoid by summing the areas of the rectangle and the two triangles.

A_trapezoid = A_rectangle + A_triangle1 + A_triangle2
A_trapezoid = a * h + 0.5 * ((b - a)/2) * h + 0.5 * ((b - a)/2) * h
A_trapezoid = a * h + ((b - a)/2) * h + ((b - a)/2) * h
A_trapezoid = a * h + (b - a) * h/2 + (b - a) * h/2
A_trapezoid = a * h + 2 * (b - a) * h/2
A_trapezoid = a * h + (b - a) * h
A_trapezoid = (a + b) * h

So, the area of the trapezoid is given by (a + b) * h.