Isosceles trapezoid

[joe]

Given an isosceles trapezoid with known height,h, bottom base, b, and top base, a<b, how to find the width any any height < h?
Thanks.

[thundre]

Given an isosceles trapezoid with known height,h, bottom base, b, and top base, a<b, how to find the width (at) any height < h?
Thanks.

It's a linear function. If x is the height where you want to measure, and y is the width at that height, the function is a line that goes through:

(0,b) -- because the bottom width is b
(h,a) -- because the top width is a

Can you find the equation for the line that goes through those two points?

Isosceles doesn't matter. This technique works for any trapezoid.

[joe]

Thank you very much for the response. I was having trouble locating a discussion of this online. I can work out the equation for the line.
I guess my next question is, how do you know it's a linear function?

[thundre]

Hmmm, I guess I know that because I skipped some steps, which I'll explain now.

Set the trapezoid on the plane such that the bottom base is along the x-axis from (0,0) to (b,0). The top base will go from ((b-a)/2,h) to (a+(b-a)/2,h). The equations for the top and bottom sides are y=0 and y=h. For the left and right sides you can work it out.

Now for any height y1 (note that I have changed the variables from my previous post), consider the line y=y1. It intersects the left and right sides at points (x1,y1) and (x2,y1). The distance between those points is x2-x1. At the bottom (y1=0), you have x1=0 and x2=b. At the top, their difference is going to be a. You can work out a formula for any point in between, and it will be linear because it's the difference of two linear functions.

[joe]

Thanks for the additional explanation. That helps a lot.