Find the ratio of circumcantre to in Radius if the ratio of sides is 4 : 5 : 7

Answer

To find the ratio of the circumradius (R) to the inradius (r) of a triangle with sides in the ratio 4:5:7, we can follow these steps:

### Step 1: Assign values to the sides
Let the sides of the triangle be:
- \( A = 4x \)
- \( B = 5x \)
- \( C = 7x \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is given by:
\[ s = \frac{A + B + C}{2} = \frac{4x + 5x + 7x}{2} = \frac{16x}{2} = 8x \]

### Step 3: Calculate the area (Δ) using Heron's formula
Using Heron's formula, the area \( Δ \) can be calculated as:
\[ Δ = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ Δ = \sqrt{8x \cdot (8x - 4x) \cdot (8x - 5x) \cdot (8x - 7x)} \]
\[ = \sqrt{8x \cdot 4x \cdot 3x \cdot x} = \sqrt{96x^4} = 4\sqrt{6}x^2 \]

### Step 4: Calculate the inradius (r)
The inradius \( r \) is given by:
\[ r = \frac{Δ}{s} = \frac{4\sqrt{6}x^2}{8x} = \frac{\sqrt{6}x}{2} \]

### Step 5: Calculate the circumradius (R)
The circumradius \( R \) is given by:
\[ R = \frac{ABC}{4Δ} \]
Calculating \( ABC \):
\[ ABC = (4x)(5x)(7x) = 140x^3 \]
Now substituting into the formula for \( R \):
\[ R = \frac{140x^3}{4 \cdot 4\sqrt{6}x^2} = \frac{140x^3}{16\sqrt{6}x^2} = \frac{35x}{4\sqrt{6}} \]

### Step 6: Find the ratio \( \frac{R}{r} \)
Now we can find the ratio of circumradius to inradius:
\[ \frac{R}{r} = \frac{\frac{35x}{4\sqrt{6}}}{\frac{\sqrt{6}x}{2}} = \frac{35x}{4\sqrt{6}} \cdot \frac{2}{\sqrt{6}x} = \frac{35 \cdot 2}{4 \cdot 6} = \frac{70}{24} = \frac{35}{12} \]

### Final Answer
Thus, the ratio of the circumradius to the inradius is:
\[ \frac{R}{r} = \frac{35}{12} \]

---