Isosceles trapezoid

Shape and space, angle and circle properties, ...
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joe
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Joined: Sat Feb 04, 2012 2:46 pm

Isosceles trapezoid

Post by joe » Sat Feb 04, 2012 2:51 pm

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
Posts: 356
Joined: Sun Mar 27, 2011 9:01 am

Re: Isosceles trapezoid

Post by thundre » Sat Feb 04, 2012 7:54 pm

joe wrote: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.
Image

joe
Posts: 3
Joined: Sat Feb 04, 2012 2:46 pm

Re: Isosceles trapezoid

Post by joe » Sun Feb 05, 2012 5:27 pm

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
Posts: 356
Joined: Sun Mar 27, 2011 9:01 am

Re: Isosceles trapezoid

Post by thundre » Mon Feb 06, 2012 2:16 am

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.
Image

joe
Posts: 3
Joined: Sat Feb 04, 2012 2:46 pm

Re: Isosceles trapezoid

Post by joe » Mon Feb 06, 2012 8:44 pm

Thanks for the additional explanation. That helps a lot.

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