Question: how do you find height in trapeziod without knowing the area. legs are 29 and bases are 15 and 57 .
Since the two legs are identical in length, then this trapezoid is symmetrical about a central axis.
Let a be the length of the top base.
Let b be the length of the bottom base.
Let c be the length of each leg.
Label the trapezoid A, B, C, D.
Because of symmetrry, we can drop a perpindicular from each end of the top base, from A and B, down to the bottom base, intersecting it at E and F.
This will give two congruent triangles at each side of the trapezoid, with the base of each triangle, DE and FC, having the same size.
EF = AB = a
DE = FC = (CD - EF)/2 = (b - a)/2
The height of the triangle is h = AE = sqrt(AD^2 - DE^2)
h = sqrt(c^2 - (b - a)^2/4)