ABC is a right angled triangle AB = 3 cm, BC = 5 cm and AC = 4 cm, then the inradius of the circle is :

Options

A. 1 cm
B. 1.25 cm
C. 1.5 cm
D. 2 cm

Answer

To find the inradius of the right-angled triangle ABC with sides AB = 3 cm, BC = 5 cm, and AC = 4 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
In triangle ABC:
- AB = 3 cm (perpendicular)
- AC = 4 cm (base)
- BC = 5 cm (hypotenuse)

### Step 2: Calculate the semi-perimeter (s) of the triangle
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{AB + AC + BC}{2} \]

Substituting the values:
\[ s = \frac{3 + 4 + 5}{2} = \frac{12}{2} = 6 \text{ cm} \]

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

### Step 4: Calculate the area (A) of the triangle
For a right-angled triangle, the area \( A \) can be calculated as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, taking AB as the height and AC as the base:
\[ A = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \text{ cm}^2 \]

### Step 5: Calculate the inradius (r)
Now substituting the values of area \( A \) and semi-perimeter \( s \) into the inradius formula:
\[ r = \frac{A}{s} = \frac{6}{6} = 1 \text{ cm} \]

### Conclusion
The inradius of triangle ABC is \( 1 \text{ cm} \).

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