Question

In a triangle \( ABC \), if \( a : b : c = 4 : 5 : 6 \), the ratio of radius of the circumcircle to that of the incircle is:

• A. \( 7:16 \)
• B. \( 17:16 \)
• C. \( 16:17 \)
• D. \( 16:7 \)

Hint:

Inradius and Circumradius Ratio}
Use \( R = \frac{abc}{4A} \), \( r = \frac{A}{s} \)
Then \( \frac{R}{r} = \frac{abc \cdot s}{4A^2} \)
Apply side ratios and compute carefully

Answer 1

The Correct Option is D

Use formula: \[ \text{Circumradius } R = \frac{abc}{4A}, \quad \text{Inradius } r = \frac{A}{s} \Rightarrow \frac{R}{r} = \frac{abc}{4A} \cdot \frac{s}{A} = \frac{abc \cdot s}{4A^2} \] After applying appropriate values for the sides and area via Heron’s formula, the ratio simplifies to \( \frac{16}{7} \).