The diagonals of an isosceles trapezoid are mutually perpendicular, find the area of the trapezoid if the point

| June 19, 2021 | education

The diagonals of an isosceles trapezoid are mutually perpendicular, find the area of the trapezoid if the point of intersection of the diagonals is 5 and 6 cm away from the bases.

Since the diagonals AC and BC intersect at right angles, the triangles BОС and AOD are rectangular.

In an isosceles trapezoid, the diagonals are whitened at the point of intersection into equal segments, ОВ = ОА, ОА = ОD, then the triangles BОС and AOD are rectangular and isosceles.

Then OK and OH are the heights, medians and bisectors of these triangles, and then the triangles BOK and AOН are also rectangular and isosceles. BK = OK = 5 cm, AH = OH = 6 cm.

Then BC = 2 * BK = 10 cm, AD = 2 * AH = 12 cm.

Height KH = OK + OH = 5 + 6 = 11 cm.

Then Savsd = (BC + AD) * KН / 2 = (10 + 12) * 11/2 = 121 cm2.

If the diagonals of an isosceles trapezoid are perpendicular, then its height is equal to the midline of the trapezoid.

Answer: The area of ​​the trapezoid is 121 cm2.



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