The length of the diagonal of the square is 4 cm. Find the radius of the circle inscribed in the square.

Given:
ABCE – square,
AC – diagonal,
AC = 4 centimeters.
Find the length of the radius of a circle inscribed in a given square, that is, r -?
Solution:
1. Consider the square ABCE. He has all sides are equal, that is, AB = BC = CB = AB.
2. Consider a right-angled triangle ABC. In it AB = BC. Let AB = BC = x centimeters. Then, by the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AB ^ 2 + BC ^ 2 = AC ^ 2:
x ^ 2 + x ^ 2 = 4 ^ 2;
x ^ 2 + x ^ 2 = 16;
2 * x ^ 2 = 16;
x ^ 2 = 16: 2;
x ^ 2 = 8;
x = 2 √ 2 centimeters.
3.r = AB: 2;
r = √ 2 centimeters.
Answer: √ 2 centimeters.