The diagonal of an isosceles trapezoid is mutually perpendicular. Find the area of the trapezoid

| February 10, 2021 | education

The diagonal of an isosceles trapezoid is mutually perpendicular. Find the area of the trapezoid if the projection of the diagonal to the larger base is 6 cm.

Since the diagonals intersect at right angles, the AMD triangle is rectangular.

Since, by condition, the trapezoid is isosceles, then its diagonals are equal, and at the point of intersection, they are divided into equal segments. AO = DO, then the triangle AOD is isosceles, and the angle DAO = ADO = 45.

Let’s draw the height of the CH. In the AСН triangle, the angle AНС is straight, the angle AСН = AСН = 450, then the triangle AСН is isosceles, AH = CH = 6 cm. And the triangle AНС, according to the Pythagorean theorem, we determine the length of the hypotenuse AС. AC ^ 2 = AH ^ 2 + CH ^ 2 = 36 + 36 = 72.

AC = √72 = 6 * √2 cm. Since the diagonals in an isosceles trapezoid are equal, then ВD = AC = 6 * √2 cm.

Let’s define the area of ​​the trapezoid through the diagonals.

S = (1/2) * AC * ВD * SinAOB = (1/2) * 6 * √2 * 6 * √2 * 1 = 36 cm2.

Answer: The area of ​​the trapezoid is 36 cm2.



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