A triangle with perimeter 7 has integer sidelengths. What is the maximum
possible area of such a triangle?

To find the maximum possible area of a triangle with integer side lengths and a perimeter of 7, let's approach the problem step-by-step.

### Step 1: Setting Up Variables
Let the side lengths of the triangle be \( a \), \( b \), and \( c \) with:
\[
a + b + c = 7
\]
and each side length \( a \), \( b \), \( c \) must be a positive integer. Additionally, the side lengths must satisfy the triangle inequality, which states:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

### Step 2: Testing Possible Combinations of Integer Side Lengths
Since \( a + b + c = 7 \), let's try values for \( a \), \( b \), and \( c \) that satisfy both the perimeter condition and the triangle inequality.

#### Possible Side Lengths
1. \( a = 1, b = 3, c = 3 \): This satisfies \( 1 + 3 + 3 = 7 \), and the triangle inequalities hold:
   - \( 1 + 3 > 3 \)
   - \( 1 + 3 > 3 \)
   - \( 3 + 3 > 1 \)

2. \( a = 2, b = 2, c = 3 \): This satisfies \( 2 + 2 + 3 = 7 \), and the triangle inequalities hold:
   - \( 2 + 2 > 3 \)
   - \( 2 + 3 > 2 \)
   - \( 2 + 3 > 2 \)

Other combinations, like \( (1, 2, 4) \), do not satisfy the triangle inequality, so the valid triangles are:
- \( (1, 3, 3) \)
- \( (2, 2, 3) \)

### Step 3: Calculating the Area Using Heron’s Formula
Heron’s formula for the area of a triangle with sides \( a \), \( b \), and \( c \) is:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( s = \frac{a + b + c}{2} \) is the semi-perimeter.

#### For \( (1, 3, 3) \):
- Semi-perimeter: \( s = \frac{1 + 3 + 3}{2} = 3.5 \)
- Area: 
  \[
  \text{Area} = \sqrt{3.5(3.5 - 1)(3.5 - 3)(3.5 - 3)} = \sqrt{3.5 \times 2.5 \times 0.5 \times 0.5} \approx 1.48
  \]

#### For \( (2, 2, 3) \):
- Semi-perimeter: \( s = \frac{2 + 2 + 3}{2} = 3.5 \)
- Area: 
  \[
  \text{Area} = \sqrt{3.5(3.5 - 2)(3.5 - 2)(3.5 - 3)} = \sqrt{3.5 \times 1.5 \times 1.5 \times 0.5} \approx 2.90
  \]

### Conclusion
The triangle with sides \( (2, 2, 3) \) has the maximum possible area of approximately \( 2.90 \) square units.

Would you like further clarification or a deeper explanation on any part of this solution?

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Here are some related questions:
1. How does Heron’s formula work, and why is it useful for finding triangle areas?
2. What are other methods for finding the area of a triangle?
3. How do the triangle inequality conditions ensure valid triangles?
4. Could we solve this problem using coordinate geometry instead?
5. What would change in this problem if the perimeter were different?

**Tip:** In optimization problems with fixed parameters (like perimeter), checking boundary conditions and constraints carefully often leads to quicker solutions.

A triangle with perimeter 7 has integer sidelengths. What is the maximum possible area of such a triangle?

Discover how to find the maximum possible area of a triangle with integer side lengths and a perimeter of 7. This solution covers the use of Heron’s formula, triangle inequalities, and optimization techniques. Perfect for students in Grades 9-10.

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