A triangle whose sides are positive integers has a perimeter 8. The area of the triangle is :

Question

Accepted Answer

To determine why Option 3 ($2\sqrt{2}$) is the correct answer for the area of a triangle with integer sides and a perimeter of 8, we need to analyze the possible side lengths and calculate the area.

### Step-by-Step Explanation:

1. **Identify Possible Side Lengths:**
   - The sides of the triangle must be positive integers, and their sum (perimeter) is 8.
   - Let's denote the sides as $a$, $b$, and $c$ such that $a + b + c = 8$.
   - Additionally, the triangle inequality must hold: $a + b > c$, $a + c > b$, and $b + c > a$.

2. **Possible Combinations:**
   - Consider the combinations of sides that satisfy both the perimeter and triangle inequality conditions:
     - $a = 3$, $b = 3$, $c = 2$
     - $a = 2$, $b = 3$, $c = 3$
     - $a = 3$, $b = 2$, $c = 3$

3. **Calculate the Area:**
   - Use Heron's formula to calculate the area of the triangle. Heron's formula is:
     $$
     \text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
     $$
     where $s$ is the semi-perimeter, $s = \frac{a+b+c}{2} = 4$.

- For the combination $a = 3$, $b = 3$, $c = 2$:
     $$
     \text{Area} = \sqrt{4(4-3)(4-3)(4-2)} = \sqrt{4 \times 1 \times 1 \times 2} = \sqrt{8} = 2\sqrt{2}
     $$

4. **Justification for Option 3:**
   - The calculated area for the triangle with sides 3, 3, and 2 is $2\sqrt{2}$, which matches Option 3.
   - This is the only combination of integer sides that satisfies the perimeter and triangle inequality conditions while yielding an area of $2\sqrt{2}$.

5. **Why Other Options are Incorrect:**
   - **Option 1 (2):** This area does not result from any valid combination of integer sides with a perimeter of 8.
   - **Option 2 ($\sqrt{2}$):** This area is not achievable with integer side lengths and the given perimeter.
   - **Option 4 ($\frac{1}{\sqrt{2}}$):** This area is not possible with the constraints provided.

In conclusion, Option 3 is the correct answer because it accurately represents the area of a triangle with integer sides and a perimeter of 8, specifically for the side lengths 3, 3, and 2.