Ideas in Geometry/Instructive examples/Proving the Area of a Trapezoid

Problem7: Proving the trapeziod area formula by picture:

Let's start by looking at the picture. We labeled the top segment b1 (base one) and the bottom segment b2(base two) (even though the original picture has it labeled as B1). The shorter segment must always be base 1, and the longer base must be base 2. We then broke up the trapezoid by drawing in the height at two places. The height "h", are drawn to create a square with base of b1, and height h. the remaining space of B2 we labeled with x on each side. x is the base for the triangles of either side of the square. The second picture shows that we took the triangle on the left side, and rotated it and moved it on top of the triangle on the right side in order to make another square. This square is of base x and height h. It wouldn't matter which side you would rotated the triangle to make the square.

We were out to prove that the area of a trapezoid is h(b1+b2)/2. First we labeled our picture in order to see what we could compare. We drew in the heights, labeling them "h" and also drew in Base 1, "B1", and "x" is the left over space in either side of b1 to make up b2. since the triangles on either side of the large rectangle are equal, we know that x is equal. and that 2x+b1=b2. We then solved for x. Knowing that B2=b1+2x, this means that b2-b1=2x. Thus x=(b2-B1)/2 Noting that the triangles were equal, we moved, and rotated the triangle on top of the other triangle to crate a rectangle with height "h" and width "x". We now have two rectangles next to each other. One of height "h" and length "b1", and the other of height "h" and length "x"

Using this logic, we then solved for the entire rectangle created. We know that the area of a rectangle is base times height. The height of the rectangle we looked for was "h" and the base was b1+x. So Area = h(b1+x). We then plugged in our previous x value (x= (b2-b1)/2) and go that area=h(b1-(b2-b1)/2). We then distributed the h value to get that area=hb1+(b2h/2)+(b1h/2). We then pulled out a 1/2 giving us that area=1/2 h (2b1+b2-b1) we then combined like terms and distributed giving us that area=(h(b1+b2))/2

Picture is Below

By: Becky Paul paul18@illinois.edu, Heather Brozowski hbrozowski@gmail.com, Tim McCormick tmccorm2@illinois.edu, Brianne Kilgallon, Sarah, and Shannon