The length of the side of the square ABCD is 4 cm. The point T lies on the side of CD and CT = 1 cm.

The length of the side of the square ABCD is 4 cm. The point T lies on the side of CD and CT = 1 cm. Calculate the length of the radius of the circle inscribed in the triangle ATD.

The ATD triangle is rectangular. By the Pythagorean theorem:

AT² = AD² + TD²; AD = 4 cm; TD = CD – CT = 4 – 1 = 3 (cm);

AT² = 4² + 3² = 16 + 9 = 25 (cm²);

AT = √25 = 5 cm;

Inscribed circle radius:

R = √ [(p – AD) (p – DT) (p – AT)] / p, where p is the semiperimeter of the triangle;

p = 1/2 (AD + DT + AT) = 1/2 (4 + 3 + 5) = 6 (cm);

R = √ [(6 – 4) (6 – 3) (6 – 5)] / 6 = √ (2 * 3 * 1) / 6 = √1 = 1 (cm).