The trapezoid diagonal ABCD divides it into 2 right-angled isosceles triangles.

The trapezoid diagonal ABCD divides it into 2 right-angled isosceles triangles. Find the midline of the trapezoid if SABC = 18 cm2

The length of the midline of the trapezoid is equal to the half-sum of the bases (BC + AD) / 2;
Consider a triangle ABC: angle B is a straight line, legs AB = BC;
We find the area of the triangle by the formula S = (AB * BC) / 2 = 18;

We express AB * BC = 18 * 2, AB * BC = 36, hence the legs AB = BC = 6 cm;

Let us find by the Pythagorean theorem a ^ 2 + b ^ 2 = c ^ 2 the hypotenuse:

AB ^ 2 + BC ^ 2 = AC ^ 2, 36 + 36 = 72;

Consider a triangle ACD: angle C – straight line, legs AC = CD, AD – hypotenuse;
Let us find by the Pythagorean theorem the hypotenuse AD: AC ^ 2 + CD ^ 2 = AD ^ 2, 72 +72 = 144;

hence AD = √144 = 12;

we find the length of the middle line (6 + 12) / 2 = 9 (cm).