A circle is inscribed in a right angled triangle of perimeter `7pi` . Then the ratio of numerical values of circumference of the circle to the area of the right angled triangle is :

Options

A. `4:7`
B. `3:7`
C. `2:7`
D. `1:7`

Answer

To solve the problem step by step, we will find the ratio of the circumference of the inscribed circle to the area of the right-angled triangle.

### Step 1: Understand the relationship between the area of the triangle and the radius of the inscribed circle.
The area \( A \) of a triangle can be expressed in terms of its perimeter \( P \) and the radius \( r \) of the inscribed circle (inradius) as follows:
\[ A = \frac{P \cdot r}{2} \]

### Step 2: Substitute the given perimeter into the area formula.
We know that the perimeter \( P \) of the triangle is given as \( 7\pi \). Substituting this into the area formula, we have:
\[ A = \frac{7\pi \cdot r}{2} \tag{1} \]

### Step 3: Write the formula for the circumference of the circle.
The circumference \( C \) of the circle is given by the formula:
\[ C = 2\pi r \tag{2} \]

### Step 4: Find the ratio of the circumference to the area.
We need to find the ratio of the circumference to the area:
\[ \text{Ratio} = \frac{C}{A} \]
Substituting equations (1) and (2) into this ratio gives:
\[ \text{Ratio} = \frac{2\pi r}{\frac{7\pi \cdot r}{2}} \]

### Step 5: Simplify the ratio.
To simplify, we can multiply by the reciprocal of the area:
\[ \text{Ratio} = \frac{2\pi r \cdot 2}{7\pi r} \]
Now, cancel \( \pi \) and \( r \) (assuming \( r \neq 0 \)):
\[ \text{Ratio} = \frac{4}{7} \]

### Final Answer:
The ratio of the numerical values of the circumference of the circle to the area of the right-angled triangle is:
\[ \frac{4}{7} \]