ABC is a right-angled triangle with AB = 6 cm and BC = 8 cm. A circle with centre O has been in scribed inside `DeltaABC`. The radius of the circle is

Options

A. 1 cm
B. 2 cm
C. 3 cm
D. 4 cm

Answer

To find the radius of the inscribed circle (inradius) of triangle ABC, we can follow these steps:

### Step 1: Identify the sides of the triangle
Given:
- AB = 6 cm (one leg of the triangle)
- BC = 8 cm (the other leg of the triangle)

Since ABC is a right-angled triangle, we can use the Pythagorean theorem to find the length of the hypotenuse AC.

### Step 2: Calculate the length of the hypotenuse AC
Using the Pythagorean theorem:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \]

### Step 3: Calculate the semi-perimeter (s) of triangle ABC
The semi-perimeter \( s \) is given by:
\[ s = \frac{AB + BC + AC}{2} = \frac{6 + 8 + 10}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 4: Calculate the area (A) of triangle ABC
The area \( A \) of a right-angled triangle can be calculated using:
\[ A = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2 \]

### Step 5: Calculate the radius (r) of the inscribed circle
The radius \( r \) of the inscribed circle can be found using the formula:
\[ r = \frac{A}{s} = \frac{24}{12} = 2 \text{ cm} \]

### Final Answer
The radius of the inscribed circle in triangle ABC is **2 cm**.

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