Prove that if in an isosceles trapezoid the diagonals are perpendicular, then the height drawn

| September 25, 2021 | education

Prove that if in an isosceles trapezoid the diagonals are perpendicular, then the height drawn to the base is equal to the midline.

Since the trapezoid ABCD is isosceles, its diagonals are equal and at point O are divided into equal segments, OB = OS, OA = OD, and since the diagonals, by condition, intersect at an angle of 90, the triangles BOC and AOD are rectangular and isosceles.

The height OH of an isosceles triangle OD is also its median and bisector, then AH = AD / 2, angle AOH = 90/2 = 45, and therefore, triangle AOH is also rectangular and isosceles, AH = OH = AD / 2.

Similarly, in the BOC triangle, OK = BK = BC / 2.

Then KН = OK + OH = BC / 2 + AD / 2 = (BC + AD) / 2, which is the middle line of the trapezoid, which was required to prove.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.