In an isosceles trapezoid, the diagonals are perpendicular. The height of the trapezoid is 16. Find its center line.

Since the trapezoid is isosceles, the diagonals AC and BD, at the point of intersection, are divided into equal segments. ОВ = ОС, ОА = ОD.

The diagonals, by condition, intersect at right angles, then the triangle AOD is rectangular and isosceles, and the angle OAD = ODA = (180 – 90) / 2 = 45.

The AOН triangle is also rectangular, and since the angle OAН = 45, the triangle is also isosceles, AH = OH = AD / 2, since OH is the height and median of the AOD triangle.

Similarly, in the triangle ВOK, OK = ВK = BC / 2.

Then KН = OK + OH = AD / 2 + BC / 2 = (AD + BC) / 2.

MР = (AD + BC) / 2 = KH = 16 cm.

Answer: The middle line of the trapezoid is 16 cm.