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

Question

Allison Allen · Accepted Answer

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#.

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#

Sophia Alfred · Answer

undefined The formula for the area of a trapezoid is A = (1/2)h(b1 + b2), where h is the height and b1 and b2 are the lengths of the two parallel sides (bases) of the trapezoid.

To find the formula for the area of a parallelogram, we can imagine splitting the parallelogram into two congruent triangles by drawing a diagonal. Each of these triangles is half of a trapezoid with bases equal to the sides of the parallelogram and height equal to the height of the parallelogram. Therefore, the area of each triangle is (1/2)h(b1 + b2), and since there are two triangles, the total area of the parallelogram is h(b1 + b2), which is the same as the formula for the area of a trapezoid when b1 = b2.

To find the formula for the area of a triangle, we can consider a triangle as a trapezoid with one of its bases having length 0. In this case, the formula for the area of the trapezoid simplifies to A = (1/2)hb, where h is the height of the triangle and b is the length of its base. This is the formula for the area of a triangle.