Finding the trapezoid of smallest perimeter given a,b,h

[uman]

Hi all,

It seems obvious that the trapezoid with the smallest perimeter given the lengths of the bases and the height is the isosceles trapezoid. How can I prove this (preferably using analytical methods)?

Note: this is not homework or anything. I need to know this as a step to prove the answer to a larger problem that I'm solving for my own pleasure, not for school.

Thanks for the help.

 

[tiny-tim]

Hi uman!

Cut out the clutter …

draw the two obvious triangles, then cut out the rectangle in between, and move the triangles together to make a new triangle …

now all you need is to minimise the perimeter of this new triangle.