A triangle (non degenerate) has integral sides and perimeter 8. if its area is A then A is:-

Options

A. less than 2
B. Greater than 2 but less than 3
C. Greater than 3 but less but 4
D. None of these

Answer

To solve the problem, we need to find the area of a non-degenerate triangle with integral sides and a perimeter of 8. Let's go through the steps systematically.

### Step 1: Understand the triangle's properties
A triangle is defined by three sides, and for it to be non-degenerate, the sum of the lengths of any two sides must be greater than the length of the third side. We also know that the perimeter of the triangle is 8, which means:

\[ a + b + c = 8 \]

where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.

### Step 2: List possible integral side combinations
Since the sides must be positive integers, we can list all combinations of \(a\), \(b\), and \(c\) that sum to 8. The combinations must also satisfy the triangle inequality:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

The possible combinations of sides are:
- \(1, 1, 6\) (not valid, as \(1 + 1 \not> 6\))
- \(1, 2, 5\) (not valid, as \(1 + 2 \not> 5\))
- \(1, 3, 4\) (not valid, as \(1 + 3 \not> 4\))
- \(2, 2, 4\) (not valid, as \(2 + 2 \not> 4\))
- \(2, 3, 3\) (valid, as \(2 + 3 > 3\))

Thus, the only valid combination is \(2, 3, 3\).

### Step 3: Calculate the semi-perimeter
The semi-perimeter \(s\) of the triangle is given by:

\[ s = \frac{a + b + c}{2} = \frac{8}{2} = 4 \]

### Step 4: Use 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 \(a = 2\), \(b = 3\), and \(c = 3\):

\[ A = \sqrt{4(4-2)(4-3)(4-3)} = \sqrt{4 \cdot 2 \cdot 1 \cdot 1} = \sqrt{4 \cdot 2} = \sqrt{8} \]

### Step 5: Simplify the area
We can simplify \(\sqrt{8}\):

\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]

### Step 6: Determine the range of the area
To find the numerical value of \(2\sqrt{2}\):

\[ \sqrt{2} \approx 1.414 \implies 2\sqrt{2} \approx 2.828 \]

### Conclusion
Thus, the area \(A\) of the triangle is approximately \(2.828\), which lies between 2 and 3.

### Final Answer
The area \(A\) is greater than 2 but less than 3.

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