Find the radius of the circle if the bases of the isosceles trapezoid described around it are 6cm and 36cm.

It is known from the condition that an isosceles trapezoid with bases of 6 cm and 36 cm is circumscribed about a circle. In order to find the radius of the inscribed circle, let’s find the lengths of the sides of the trapezoid.

So, in the described quadrangle, the sums of the opposite sides are equal.

Let’s find the side. We denote by the variable x – the lateral side of the trapezoid, a and b – the base (a = 36, b = 6), then:

a + b = 2x;

42 = 2x;

x = 21 cm.

The diameter of the inscribed circle is equal to the height of the trapezoid.

Let us lower the trapezoid heights BK and CM from the vertices B and C to the base AD = a = 36 cm.

And consider a right-angled triangle ABK:

AK = (a – b) / 2 = 15 cm.

We apply the Pythagorean theorem and get:

BK = √ (c ^ 2 – AK ^ 2) = √ (441 – 225) = √216 = 6√6 cm diameter length.

Then the radius is:

r = 3√6 cm.