Find the area of an isosceles trapezoid whose diagonals are perpendicular and the sum of the bases is 20 cm.
September 4, 2021 | education
| Draw the height HK through the point of intersection of the diagonals.
Since the trapezoid is isosceles, its diagonals at point O are divided into equal segments.
ОВ = ОС, ОА = ОD, then triangles BОС and AOD are rectangular and isosceles.
OH and OK heights, medians and bisectors of triangles BOC and AOD, then in right-angled triangles BON and AOK the acute angles are 45, and then the triangles are right-angled and isosceles. OH = BH = BC / 2, OK = AK = AD / 2.
Then HK = BC / 2 + AD / 2 = (BC + AD) / 2.
By condition, (ВС + АD) = 20 cm, then КН = 20/2 = 10 cm.
Savsd = (ВС + АD) * КH / 2 = 20 * 10/2 = 100 cm2.
Answer: The area of the trapezoid is 100 cm2.
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