Find the ratio of circumradius to inradius . If the ratio of sides is `5:7:8`.

Answer

To find the ratio of the circumradius (R) to the inradius (r) of a triangle with sides in the ratio 5:7:8, we can follow these steps:

### Step 1: Define the sides of the triangle
Let the sides of the triangle be:
- \( a = 5x \)
- \( b = 7x \)
- \( c = 8x \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ s = \frac{5x + 7x + 8x}{2} = \frac{20x}{2} = 10x \]

### Step 3: Calculate the area (Δ) of the triangle
Using Heron's formula, the area \( Δ \) can be calculated as:
\[ Δ = \sqrt{s(s - a)(s - b)(s - c)} \]
Calculating each term:
- \( s - a = 10x - 5x = 5x \)
- \( s - b = 10x - 7x = 3x \)
- \( s - c = 10x - 8x = 2x \)

Now substituting these values into the area formula:
\[ Δ = \sqrt{10x \cdot 5x \cdot 3x \cdot 2x} = \sqrt{300x^4} = 10\sqrt{3}x^2 \]

### Step 4: Calculate the circumradius (R)
The circumradius \( R \) is given by the formula:
\[ R = \frac{abc}{4Δ} \]
Substituting the values of \( a \), \( b \), \( c \), and \( Δ \):
\[ R = \frac{(5x)(7x)(8x)}{4(10\sqrt{3}x^2)} = \frac{280x^3}{40\sqrt{3}x^2} = \frac{7x}{\sqrt{3}} \]

### Step 5: Calculate the inradius (r)
The inradius \( r \) is given by the formula:
\[ r = \frac{Δ}{s} \]
Substituting the values of \( Δ \) and \( s \):
\[ r = \frac{10\sqrt{3}x^2}{10x} = \sqrt{3}x \]

### Step 6: Calculate the ratio \( \frac{R}{r} \)
Now we can find the ratio of the circumradius to the inradius:
\[ \frac{R}{r} = \frac{\frac{7x}{\sqrt{3}}}{\sqrt{3}x} = \frac{7}{3} \]

### Final Answer
The ratio of the circumradius to the inradius is:
\[ \frac{R}{r} = \frac{7}{3} \]