​Circles are inscribed and circumscribed to a triangle whose sides are 3 cm, 4 cm and 5 cm. What is the ratio of radius of the in-circle to that of the circumcircle?​

Correct option is C

Given:

Sides of the triangle: 3 cm, 4 cm, and 5 cm

Formula Used:

Inradius of a right-angled triangle:

​$$r = \frac{(a + b - c) } {2}$$

where a, b, and c are the sides of the triangle.

​Circumradius of a right-angled triangle:

​$$R =\frac{c }{ 2}$$

where c is the hypotenuse of the triangle.​​

Solution:

The given triangle with side 3cm, 4cm, 5cm is a right-angled triangle (Pythagorean triplet) 

Calculating Inradius (r):

​$$r = \frac{(3 + 4 - 5) }{ 2 }= 1 cm$$​​

Calculating Circumradius (R):

​$$R =\frac {5 }{2} = 2.5 cm$$​​

Calculating the Ratio:

​$$\frac {r }{R} =\frac{ 1 cm} { 2.5 cm} =$$ 2 : 5

Therefore, the ratio of the radius of the in-circle to that of the circumcircle is 2 : 5.

Option (c) is right answer.