Let triangle ABC is right triangle right angled at C such that `A lt B` and `r=8, R=41` .
Q. Area of `DeltaABC` is :

Options

A. 720
B. 1440
C. 360
D. 480

Answer

To find the area of triangle ABC, which is a right triangle right-angled at C, we can use the relationship between the circumradius (R), inradius (r), and the area (Δ) of the triangle.

Given:
- r = 8 (inradius)
- R = 41 (circumradius)

### Step-by-Step Solution:

1. **Understanding the Relationship**:
In a right triangle, the area can be expressed using the circumradius (R) and inradius (r) with the formula:
\[ \Delta = r \cdot s \]
where \(s\) is the semi-perimeter of the triangle.

2. **Finding the Semi-Perimeter**:
The semi-perimeter \(s\) can also be expressed in terms of the circumradius:
\[ s = \frac{a + b + c}{2} \]
where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse.

3. **Using the Right Triangle Properties**:
For a right triangle, we know:
\[ c = 2R \]
Thus, substituting the value of R:
\[ c = 2 \times 41 = 82 \]

4. **Applying the Relationship**:
The sum of the lengths of the two legs \(a + b\) can be expressed as:
\[ a + b = 2R + 2r \]
Substituting the known values:
\[ a + b = 2 \times 41 + 2 \times 8 = 82 + 16 = 98 \]

5. **Calculating the Semi-Perimeter**:
Now, substituting into the semi-perimeter formula:
\[ s = \frac{a + b + c}{2} = \frac{98 + 82}{2} = \frac{180}{2} = 90 \]

6. **Finding the Area**:
Now, substituting the values of \(r\) and \(s\) into the area formula:
\[ \Delta = r \cdot s = 8 \cdot 90 = 720 \]

### Conclusion:
The area of triangle ABC is \(720\) square units.