Question

three circles of radii 1, 2,3 units touch each other externally in the plane.What is the circum radius of the triangle formed by joining the centers of the circles?

Answer 1

circumradius formula^
a,b and c are the lengths of the sides of the triangle formed by joining the centres of the circles

a=3
b=4
c=5

Answer 2

The triangle formed will have the sides :

(1 + 2) , (2 + 3) , (1 + 3) units

⇒ 3 , 4 , 5 units .

Shame on me , the formula was given in MTRP [ the greatest book of maths I have come across in class 9 ] Class 9 .

Circum radius of triangle = abc / ( 4 A )

Here a , b , c are the sides of triangle and A is the area of the triangle .

Area of the triangle will be :

√[ s(s - a)(s - b)(s - c) ]

Thats Heron's formula ↑

s is the semi-perimeter .

s = (a + b + c)/2

⇒ s = (3 + 4 + 5)/2

⇒ s = (8 + 4)/2

⇒ s = 12/2

⇒ s = 6

Area = √[ 6 × (6 - 3) × (6 - 4) × (6 - 5) ]

⇒ √[ 6 × 3 × 2 × 1 ]

⇒ √[ 6 × 6 ]

⇒ 6

Circum radius = (3 × 4 × 5) / ( 4 × 6 )

⇒ ( 3 × 5 )/6

⇒ 5/2

⇒ 2.5

The circum radius of the triangle will be 2.5 units !