A circle is described around the square, and a circle is inscribed in the square.

A circle is described around the square, and a circle is inscribed in the square. Find the radius of the inscribed circle if the radius of the inscribed circle is 10√2 cm.

1. You can notice that on the circumscribed circle there are the vertices of the square, so the diagonal of the square is equal to the diameter of this circle:

Do = 2 * Ro = C = 2 * 10 * √ (2) cm;

2. The inscribed circle touches the midpoints of the sides of the square, its diameter is equal to the length of the side of the square:

Db = 2 * Rb = A cm;

3. According to our favorite Pythagorean theorem:

C² = A² + B² = 2 * A² (a square is a square because it has equal sides);

A = √ (C² / 2) = √ ((2 * 10 * √ (2)) ² / 2) = 20 cm;

4. Inscribed circle radius: Rb cm;

Rb = Db / 2 = 20/2 = 10 cm.

Answer: Rb = 10 cm.