ABC is a right triangle in which `angle B = 90^@`. If `AB = 8" cm" and BC = 6" cm"`. find the diameter of the circle inscribed in the triangle.

Answer

To find the diameter of the circle inscribed in triangle ABC, where angle B is 90 degrees, AB = 8 cm, and BC = 6 cm, we follow these steps:

### Step 1: Identify the sides of the triangle
In triangle ABC:
- AB = 8 cm (one leg)
- BC = 6 cm (the other leg)
- AC = hypotenuse (to be calculated)

### Step 2: Calculate the length of the hypotenuse AC using the Pythagorean theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values:
\[ AC^2 = 8^2 + 6^2 = 64 + 36 = 100 \]
Taking the square root:
\[ AC = \sqrt{100} = 10 \text{ cm} \]

### Step 3: Calculate the area of triangle ABC
The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take AB as the base and BC as the height:
\[ A = \frac{1}{2} \times 8 \times 6 = \frac{48}{2} = 24 \text{ cm}^2 \]

### Step 4: Calculate the semi-perimeter of triangle ABC
The semi-perimeter \( s \) is given by:
\[ s = \frac{AB + BC + AC}{2} \]
Substituting the values:
\[ s = \frac{8 + 6 + 10}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 5: Calculate the radius \( R \) of the inscribed circle
The radius \( R \) of the inscribed circle can be calculated using the formula:
\[ R = \frac{A}{s} \]
Substituting the area and semi-perimeter:
\[ R = \frac{24}{12} = 2 \text{ cm} \]

### Step 6: Calculate the diameter of the inscribed circle
The diameter \( d \) is given by:
\[ d = 2R \]
Substituting the value of \( R \):
\[ d = 2 \times 2 = 4 \text{ cm} \]

### Final Answer
The diameter of the circle inscribed in triangle ABC is **4 cm**.

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