The area of a right triangle is 6864 sq. units. If the ratio of its legs is `143 : 24`, then the value of r is ______

Explanation

To solve the problem step by step, we will follow the given information about the right triangle and use the properties of triangles.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that the area of the right triangle is 6864 sq. units and the ratio of its legs (a and b) is 143:24.

2. **Setting Up the Ratio:**
Let the lengths of the legs of the triangle be represented as:
\[ a = 143k \quad \text{and} \quad b = 24k \]
where \( k \) is a constant.

3. **Finding the Area Formula:**
The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times a \times b \]
Substituting the values of \( a \) and \( b \):
\[ A = \frac{1}{2} \times (143k) \times (24k) = \frac{1}{2} \times 3432k^2 = 1716k^2 \]

4. **Setting Up the Equation:**
Since the area is given as 6864 sq. units, we can set up the equation:
\[ 1716k^2 = 6864 \]

5. **Solving for \( k^2 \):**
To find \( k^2 \), we divide both sides by 1716:
\[ k^2 = \frac{6864}{1716} \]
Simplifying this gives:
\[ k^2 = 4 \]

6. **Finding \( k \):**
Taking the square root of both sides:
\[ k = 2 \]

7. **Finding the Lengths of the Legs:**
Now we can find the lengths of the legs \( a \) and \( b \):
\[ a = 143k = 143 \times 2 = 286 \]
\[ b = 24k = 24 \times 2 = 48 \]

8. **Finding the Hypotenuse:**
Using the Pythagorean theorem to find the hypotenuse \( c \):
\[ c^2 = a^2 + b^2 = 286^2 + 48^2 \]
Calculating \( c^2 \):
\[ c^2 = 81796 + 2304 = 84000 \]
Taking the square root gives:
\[ c = \sqrt{84000} = 290 \]

9. **Calculating the Semi-Perimeter \( s \):**
The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} = \frac{286 + 48 + 290}{2} = \frac{624}{2} = 312 \]

10. **Finding the Inradius \( r \):**
The inradius \( r \) can be calculated using the formula:
\[ r = \frac{A}{s} = \frac{6864}{312} \]
Simplifying this gives:
\[ r = 22 \]

### Final Answer:
The value of \( r \) is \( \boxed{22} \).