Question

in a triangle abc, a : b : c = 4 : 5 : 6. then ratio of the radius of the circumcircle to that of the incircle is

Answer 1

Answer :.

The ratio of radius of circumcircle to that of incircle is 16/7

Step-by-step Explanation :

Given : a : b : c = 4 : 5 : 6

To find : ratio of the radius of the circumcircle to that of the incircle = ?

Let a = 4k. , b = 5k , c = 6k

Therefore : s = 15/2 k , s - a = 7/2 k , s - b = 5/2 k and s - c = 3/2 k

∆² = s( s - a )( s - b )( s - c )

∆² = 15 × 7 × 5 × 3 × (k/2)⁴

Or ∆ = 15√7 × (k/2)²

Also,

Inradius:

r = ∆ / s

= 15√7 × (k/2)² / 15/2 k

= √7 k/2

Circumradius :

R = abc/ 4∆

= 4 × 5 × 6 × k³ / 4 × 15√7 × (k²/4)

= 8/√7 × k

∴ R/r = 16/7

Hence the ratio is 16/7