Consider a right triangle with legs of length `a` and `b` and hypotenuse of length` c`. If `k` denotes the area of the triangle then the value of ` (a+b) ` equals

Options

A. `c+2sqrt(k)`
B. `sqrt(c^(2) + 4k)`
C. `(c^(2))/(4sqrt(k))`
D. `(4k)/(c )`

Answer

To solve the problem, we need to find the value of \( a + b \) in terms of \( c \) (the hypotenuse) and \( k \) (the area of the triangle).

### Step-by-Step Solution:

1. **Understand the Right Triangle**:
We have a right triangle with legs \( a \) and \( b \), and the hypotenuse \( c \).

2. **Apply the Pythagorean Theorem**:
According to the Pythagorean theorem, we have:
\[ c^2 = a^2 + b^2 \]

3. **Express the Area of the Triangle**:
The area \( k \) of the triangle can be expressed as:
\[ k = \frac{1}{2} \times a \times b \]
Rearranging gives:
\[ ab = 2k \]

4. **Use the Identity for \( (a + b)^2 \)**:
We know the identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Substituting \( a^2 + b^2 \) and \( ab \) into this identity, we get:
\[ (a + b)^2 = c^2 + 2(2k) = c^2 + 4k \]

5. **Take the Square Root**:
To find \( a + b \), we take the square root of both sides:
\[ a + b = \sqrt{c^2 + 4k} \]

### Final Result:
Thus, the value of \( a + b \) is:
\[ \boxed{\sqrt{c^2 + 4k}} \]