We notice that if we'll add the squares of the lengths of 2 sides of the triangle ABC, we'll get the square of the biggest one.
5^2 = 3^2 + 4^2
25 = 9 + 16
25 = 25
This equality certifies the fact that the triangle ABC is a right angled triangle, whose right angle is A = 90 degrees.
The opposite side to the right angle A is called hypothenuse, BC.
The hypothenuse represents the diameter of the circle and we'll get:
R = hypotenuse/2
Since hypotenuse is the biggest side of a right angle triangle, that means that BC = 5.
R = 5/2
The radius of the inscribed circle in the given triangle is:
r = S/p
S = AB*AC/2
S = 3*4/2
S = 6 square units (area of the triangle ABC)
p = (3+4+5)/2
p = 12/2
p = 6 units
r = 6 square units/ 6 units
r = 1 unit.
The radius of the inscribed circle is r = 1 unit.