Question

Explain how the formula for the area of a trapezoid can be used to find the formulas for the areas of parallelograms and triangles?

Answer 1

Trapezoid has bases `a` and `b`, and the altitude `h`.
Its area is `(a+b)/2*h`.
Parallelogram: `a=b`, `=>` area is `(a+a)/2*h=a*h`
Triangle: `b=0`, `=>` area is `1/2*a*h`.

Explanation:

Consider a trapezoid with bases `a` and `b` and an altitude `h`.

If `a=b`, trapezoid becomes a parallelogram because any quadrilateral with two opposite side congruent and parallel to each other is parallelogram.
So, the area of parallelogram can be obtained using a formula for an area of parallelogram, taking into consideration `a=b`, which result in
`S_p = (a+a)/2*h=a*h`

If `b=0`, trapezoid becomes a triangle because of side will have zero length and quadrilateral turns into triangle.
So, the area of triangle can be obtained using a formula for an area of parallelogram, taking into consideration `b=0`, which result in
`S_t = a/2*h=1/2*a*h`