In equilateral triangle , ratio of inradius ,circum radii and exadii is :-

Options

A. `2:3:5`
B. `1:2:3`
C. `3:7:9`
D. `3:7: 8`

Answer

To find the ratio of the inradius (r), circumradius (R), and exradius (r1) of an equilateral triangle, we will follow these steps:

### Step 1: Define the parameters of the equilateral triangle
Let the side length of the equilateral triangle be \( a \). In an equilateral triangle, all sides are equal, and all angles are \( 60^\circ \).

### Step 2: Calculate the area (Δ) of the equilateral triangle
The area \( Δ \) of an equilateral triangle can be calculated using the formula:
\[ Δ = \frac{\sqrt{3}}{4} a^2 \]

### Step 3: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is given by:
\[ s = \frac{a + a + a}{2} = \frac{3a}{2} \]

### Step 4: Calculate the inradius (r)
The inradius \( r \) can be calculated using the formula:
\[ r = \frac{Δ}{s} \]
Substituting the values of \( Δ \) and \( s \):
\[ r = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2}} = \frac{\sqrt{3} a^2}{4} \cdot \frac{2}{3a} = \frac{\sqrt{3} a}{6} \]

### Step 5: Calculate the circumradius (R)
The circumradius \( R \) can be calculated using the formula:
\[ R = \frac{abc}{4Δ} \]
Since \( a = b = c \), we have:
\[ R = \frac{a \cdot a \cdot a}{4 \cdot \frac{\sqrt{3}}{4} a^2} = \frac{a^3}{\sqrt{3} a^2} = \frac{a}{\sqrt{3}} \]

### Step 6: Calculate the exradius (r1)
The exradius \( r1 \) can be calculated using the formula:
\[ r1 = \frac{Δ}{s - a} \]
Substituting the values of \( Δ \) and \( s \):
\[ r1 = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{3a}{2} - a} = \frac{\frac{\sqrt{3}}{4} a^2}{\frac{a}{2}} = \frac{\sqrt{3} a^2}{4} \cdot \frac{2}{a} = \frac{\sqrt{3} a}{2} \]

### Step 7: Find the ratio of inradius, circumradius, and exradius
Now we have:
- \( r = \frac{\sqrt{3}}{6} a \)
- \( R = \frac{a}{\sqrt{3}} \)
- \( r1 = \frac{\sqrt{3}}{2} a \)

The ratio \( r : R : r1 \) is:
\[ \frac{\sqrt{3}}{6} a : \frac{a}{\sqrt{3}} : \frac{\sqrt{3}}{2} a \]

Dividing each term by \( a \):
\[ \frac{\sqrt{3}}{6} : \frac{1}{\sqrt{3}} : \frac{\sqrt{3}}{2} \]

To simplify, we can multiply through by \( 6\sqrt{3} \):
\[ \sqrt{3} : 2 : 9 \]

Thus, the final ratio of the inradius, circumradius, and exradius of an equilateral triangle is:
\[ \sqrt{3} : 2 : 9 \]