Find the area of an isosceles trapezoid, knowing that the diagonals are mutually
September 28, 2021 | education
| Find the area of an isosceles trapezoid, knowing that the diagonals are mutually perpendicular and their lengths are 16 cm.
The perpendicularity of the diagonals forms two similar right-angled isosceles triangles in an isosceles trapezium, in which the bases of the trapezoid are the hypotenuses.
Let the diagonals be divided by the point of intersection into segments m and (16 – m).
From the Pythagorean theorems, we obtain an expression for the sum of the bases of a trapezoid:
a + b = (2 * x ^ 2) ^ (1/2) + (2 * (16 – x) ^ 2) ^ (1/2) = x * 2 ^ (1/2) + (16 – x ) * 2 ^ (1/2) = 16 * 2 ^ (1/2);
The height in such a trapezoid is equal to half the sum of the bases:
h = 8 * 2 ^ (1/2);
The area is equal to:
S = 1/2 * 16 * 8 * 2 = 128 cm².
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