Question:
In a \(\Delta ABC\) if \( a:b:c = 5:6:7 \), then the ratio of the radius of the circumcircle to that of the incircle is:
In a \(\Delta ABC\) if \( a:b:c = 5:6:7 \), then the ratio of the radius of the circumcircle to that of the incircle is:
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Memorize the formula \( R/r = \frac{abc}{4(s-a)(s-b)(s-c)} \), which is very fast when side ratios are given as integers.
Updated On: Jun 9, 2026
- \( 35:16 \)
- \( 7:5 \)
- \( 9:7 \)
- \( 16:9 \)
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The Correct Option is A
Solution and Explanation
Concept:
The ratio of circumradius \( R \) to inradius \( r \) is given by \( R/r = \frac{abc}{4\Delta} / \frac{\Delta}{s} = \frac{abc \cdot s}{4\Delta^2} \). Using Heron's formula \(\Delta = \sqrt{s(s-a)(s-b)(s-c)}\), the ratio simplifies to \( R/r = \frac{abc}{(a+b+c)(b+c-a)(a+c-b)(a+b-c)} \times \dots \) which is often calculated via \( \frac{1}{4\sin(A/2)\sin(B/2)\sin(C/2)} \).
Step 1: Define sides \( a=5k, b=6k, c=7k \).
\( s = (5+6+7)k / 2 = 9k \). Area \( \Delta = \sqrt{9k(4k)(3k)(2k)} = \sqrt{216k^4} = 6\sqrt{6}k^2 \).
Step 2: Calculate \( R \) and \( r \).
\( R = \frac{abc}{4\Delta} = \frac{210k^3}{4(6\sqrt{6}k^2)} = \frac{210k}{24\sqrt{6}} = \frac{35k}{4\sqrt{6}} \). \( r = \frac{\Delta}{s} = \frac{6\sqrt{6}k^2}{9k} = \frac{2\sqrt{6}k}{3} \).
Step 3: Calculate ratio \( R/r \).
\( \frac{R}{r} = \frac{35k}{4\sqrt{6}} \cdot \frac{3}{2\sqrt{6}k} = \frac{105}{4 \cdot 6} = \frac{105}{24} = \frac{35}{8} \). Correction: \( R/r = 35/16 \) (Check: \( 8 \times 2 \)? \( 4 \times 6 = 24 \). \( 105/24 = 35/8 \). Let's recompute). \( R = 35k/4\sqrt{6} \), \( r = 2\sqrt{6}k/3 \). \( R/r = (35 \cdot 3) / (4 \cdot 6 \cdot 2) = 105/48 = 35/16 \). 35:16
Step 1: Define sides \( a=5k, b=6k, c=7k \).
\( s = (5+6+7)k / 2 = 9k \). Area \( \Delta = \sqrt{9k(4k)(3k)(2k)} = \sqrt{216k^4} = 6\sqrt{6}k^2 \).
Step 2: Calculate \( R \) and \( r \).
\( R = \frac{abc}{4\Delta} = \frac{210k^3}{4(6\sqrt{6}k^2)} = \frac{210k}{24\sqrt{6}} = \frac{35k}{4\sqrt{6}} \). \( r = \frac{\Delta}{s} = \frac{6\sqrt{6}k^2}{9k} = \frac{2\sqrt{6}k}{3} \).
Step 3: Calculate ratio \( R/r \).
\( \frac{R}{r} = \frac{35k}{4\sqrt{6}} \cdot \frac{3}{2\sqrt{6}k} = \frac{105}{4 \cdot 6} = \frac{105}{24} = \frac{35}{8} \). Correction: \( R/r = 35/16 \) (Check: \( 8 \times 2 \)? \( 4 \times 6 = 24 \). \( 105/24 = 35/8 \). Let's recompute). \( R = 35k/4\sqrt{6} \), \( r = 2\sqrt{6}k/3 \). \( R/r = (35 \cdot 3) / (4 \cdot 6 \cdot 2) = 105/48 = 35/16 \). 35:16
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