Question

Three circles of radii 2, 4, and 6 units respectively touch each other externally in the plane. The circumradius of the
triangle formed by joining the centers of the circles is!​

Answer 1

Answer:

★ Circumradius = 5 units ★

Step-by-step explanation:

Given:

• Radii of three circles are 2,4 and 6 units respectively.

To Find:

• Circumference of the triangle formed by joining centers of circles is

Solution: The triangle will form by joining these radii will have following sides:

• (2+4) , (4+6) , (2+6)
• 6 , 10 and 8 units.

★ Circumradius of triangle= (abc/4A) ★

• a , b & c are sides of triangle and A is area of triangle

★ Area of triangle = √S(s–a)(s–b)(s–c) (By Heron's formula)

† Where S = Semi perimeter †

• Formula for semi perimeter ↓

→ S = (a + b + c/2)

$\small\implies{\sf }$ S = ( 6 + 10 + 8/2)

$\small\implies{\sf }$ S = 24/2 = 12 units

Now area of Triangle:-

$\small\implies{\sf }$ √S(s–a)(s–b)(s–c)

$\small\implies{\sf }$ √12(12–6)(12–10)(12–8)

$\small\implies{\sf }$ √12 x 6 x 2 x 4

$\small\implies{\sf }$ √2 x 2 x 3 x 2 x 3 x 2 x 2 x 2

$\small\implies{\sf }$ 2 x 2 x 2 x 3

$\small\implies{\sf }$ 24 sq units

Now Putting values on formula of Circumradius :

$\small\implies{\sf }$ (abc/4A)

$\small\implies{\sf }$ ( 6 x 10 x 8/4 x 24)

$\small\implies{\sf }$ 480/96

$\small\implies{\sf }$ 5

Hence, Circumradius of triangle will be 5 units.