The large side of the trapezoid is 42 cm. The inscribed circle divides one of the lateral

The large side of the trapezoid is 42 cm. The inscribed circle divides one of the lateral sides into segments of 8 and 18 cm. Find the area of the trapezoid.

The circle is divided by points of tangency:
a larger base on segments of 18 cm and (42 – 18) = 24 cm;
smaller base for 8 cm and x cm segments;
the other side into segments 24 cm and x cm.

If we draw the heights of the trapezoid, then they will cut off two right-angled triangles. Consider the one for which the hypotenuse is known (lateral side 8 + 18 = 26 (cm)), and legs 18 – 8 = 10 (cm) and unknown (trapezoid height) – h cm.By the Pythagorean theorem:

h ^ 2 = 26 ^ 2 – 10 ^ 2 = 676 – 100 = 576;

h = 24 cm.

Now, in another right-angled triangle, the hypotenuse is (24 + x) cm, one leg (height) is 24 cm, and the other is (24 – x) cm.

(24 + x) ^ 2 = (24 – x) ^ 2 + 24 ^ 2;

96x = 576;

x = 6.

So the upper base is 8 + 6 = 14 (cm), and the area is:

((14 + 42) / 2) * 24 = 672 (sq. Cm).