A triangle with integral sides has perimeter 8 cm. Then find the area of the triangle

Explanation

To find the area of a triangle with integral sides and a perimeter of 8 cm, we can follow these steps:

### Step 1: Define the sides of the triangle
Let the sides of the triangle be \( a \), \( b \), and \( c \). According to the problem, we have:
\[ a + b + c = 8 \]

### Step 2: Apply the triangle inequality
For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we need to satisfy the following inequalities:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

### Step 3: Find integral combinations of sides
We can use trial and error to find combinations of \( a \), \( b \), and \( c \) that satisfy both the perimeter condition and the triangle inequality.

Let's consider possible combinations:
- If \( a = 3 \), \( b = 3 \), then \( c = 8 - 3 - 3 = 2 \).
- Check the triangle inequalities:
- \( 3 + 3 > 2 \) (True)
- \( 3 + 2 > 3 \) (True)
- \( 3 + 2 > 3 \) (True)

This combination \( (3, 3, 2) \) satisfies all conditions.

### Step 4: Calculate the semi-perimeter
Now, we calculate the semi-perimeter \( s \):
\[ s = \frac{a + b + c}{2} = \frac{8}{2} = 4 \]

### Step 5: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
- \( s = 4 \)
- \( a = 3 \)
- \( b = 3 \)
- \( c = 2 \)

We calculate:
\[ A = \sqrt{4(4-3)(4-3)(4-2)} = \sqrt{4 \times 1 \times 1 \times 2} = \sqrt{8} = 2\sqrt{2} \]

### Final Answer
The area of the triangle is:
\[ \text{Area} = 2\sqrt{2} \text{ cm}^2 \]

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