The bases of an isosceles trapezoid circumscribed about a circle are 4cm and 16cm. Find the side and height of the trapezoid.

The sums of the lengths of the opposite sides of a quadrilateral circumscribed about a circle are equal. This means that the sum of the lengths of the bases of this isosceles trapezoid is equal to the sum of the lengths of its lateral sides. Accordingly, each side is equal to the half-sum of the bases:

(4 + 16) / 2 = 10 cm.

The larger base of an isosceles trapezoid is equal to the sum of the lengths of the smaller base and the projections of the two lateral sides onto the larger base. Find the side projection:

(16 – 4) / 2 = 12/2 = 6 cm.

The height of the trapezoid, the side and its projection form a right-angled triangle, from which, according to the Pythagorean theorem, we find the height:

h ^ 2 = 10 ^ 2 – 6 ^ 2 = 100 – 36 = 64 = 82;

h = 8 cm.