The sides of a right triangle ABC are a, b and c, where c is the hypotenuse. What will be the radius of the in circle of this triangle?

Options

A. `((a+b+c))/2`
B. `((a+b-c))/2`
C. `((b+c-a))/2`
D. `((a+c-b))/2`

Answer

To find the radius of the incircle of a right triangle ABC, where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \( a \) and \( b \) be the lengths of the two legs of the right triangle.
- Let \( c \) be the length of the hypotenuse.

2. **Recall the formula for the radius of the incircle**:
- The radius \( r \) of the incircle of a triangle can be calculated using the formula:
\[ r = \frac{a + b - c}{2} \]
- This formula is specific to right triangles and is derived from the area and semiperimeter of the triangle.

3. **Substitute the values**:
- Substitute the values of \( a \), \( b \), and \( c \) into the formula to find \( r \).

4. **Calculate the radius**:
- Perform the arithmetic operation to find the value of \( r \).

### Example Calculation:
- If \( a = 3 \), \( b = 4 \), and \( c = 5 \):
\[ r = \frac{3 + 4 - 5}{2} = \frac{2}{2} = 1 \]

### Final Answer:
- The radius of the incircle \( r \) is given by the formula \( r = \frac{a + b - c}{2} \).