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

| May 10, 2021 | education

The first way.

If in an isosceles trapezoid the diagonals intersect at right angles, then the middle line of the trapezoid is equal to its height. MР = KН = 16 cm.

Second way.

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

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

The AOH triangle is also rectangular, and since the angle OAH = 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 BOK, OK = BK = BC / 2.

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

MR = (AD + BC) / 2 = KH = 16 cm.

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



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