Question

If the lengths of three sides of a triangle are 6 cm, 8 cm and 10 cm, then the radius of the circumcircle of the triangle is
1. 5 cm
2. 6 cm
3. 7 cm
4. None of the above

Answer 1

Correct Answer - Option 1 : 5 cm

Given:

Sides of triangle = 8 cm, 6 cm, 10 cm

Concept used: 

In the right triangle, the hypotenuse is the diameter of the circumcircle, and its centre is exactly at the midpoint of the hypotenuse.

This is the same situation as Thales Theorem, where the diameter subtends a right angle to any point on a circle's circumference.

Formulae required:

In rightangle triangle a² + b² = c²

Calculations:

If given sides

a = 8 cm 

b = 6 cm

c = 10 cm 

Satisfies the above formula it is right-angle triangle 

8² + 6² = 64 + 36

⇒ 100

10² = 100

Hence the sides satisfied the condition so this is a right-angle triangle

In the right-angle triangle, the longest side is the hypotenuse

Hence, hypotenuse = 10 cm

According to the concept, 

The diameter of circumcircle = hypotenuse

Diameter = 10 cm

Radius = diameter/2

⇒ 10/2 = 5 cm

∴ The radius of the circumcircle of a triangle for the given sides is 5 cm 

6, 8, and 10 are the Pythagoras triplets so, the triangle is a right-angled triangle 

The midpoint of hypotenuse is the midpoint of the diameter