In a triangle ABC right angled at B, the inradius r is equal to

Options

A. `c+b-a`
B. `a+b+c`
C. `(a+c-b)/2`
D. none

Answer

To find the inradius \( r \) of a triangle \( ABC \) that is right-angled at \( B \), we can follow these steps:

### Step 1: Understand the Triangle
We have a right-angled triangle \( ABC \) with the right angle at \( B \). Let's denote the lengths of the sides as follows:
- \( AB = c \) (the side opposite to angle \( C \))
- \( BC = a \) (the side opposite to angle \( A \))
- \( AC = b \) (the hypotenuse)

### Step 2: Calculate the Semi-perimeter \( s \)
The semi-perimeter \( s \) of triangle \( ABC \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]

### Step 3: Use the Formula for Inradius
The inradius \( r \) of a triangle can be calculated using the formula:
\[ r = \frac{A}{s} \]
where \( A \) is the area of the triangle.

### Step 4: Calculate the Area \( A \)
For a right-angled triangle, the area \( A \) can be calculated as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times c \]

### Step 5: Substitute \( A \) and \( s \) into the Inradius Formula
Now we substitute the values of \( A \) and \( s \) into the inradius formula:
\[ r = \frac{\frac{1}{2} \times a \times c}{\frac{a + b + c}{2}} \]

### Step 6: Simplify the Expression
This simplifies to:
\[ r = \frac{a \times c}{a + b + c} \]

### Step 7: Final Expression
Thus, the inradius \( r \) of triangle \( ABC \) right-angled at \( B \) is:
\[ r = \frac{a \times c}{a + b + c} \]