Isosceles Trapezoid - Diagonals and Height

Diagonals and Height

The diagonals of an isosceles trapezoid have the same length; that is, every isosceles trapezoid is an equidiagonal quadrilateral. Moreover, the diagonals divide each other in the same proportions. As pictured, the diagonals AC and BD have the same length (AC = BD) and divide each other into segments of the same length (AE = DE and BE = CE).

The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is,

The length of each diagonal is, according to Ptolemy's theorem, given by

where a and b are the lengths of the parallel sides AD and BC, and c is the length of each leg AB and CD.

The height is, according to the Pythagorean theorem, given by

The distance from point E to base AD is given by

where a and b are the lengths of the parallel sides AD and BC, and h is the height of the trapezoid.