Math Problem Statement

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

Solution

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 aa, bb, and cc with: a+b+c=7a + b + c = 7 and each side length aa, bb, cc must be a positive integer. Additionally, the side lengths must satisfy the triangle inequality, which states:

  1. a+b>ca + b > c
  2. a+c>ba + c > b
  3. b+c>ab + c > a

Step 2: Testing Possible Combinations of Integer Side Lengths

Since a+b+c=7a + b + c = 7, let's try values for aa, bb, and cc that satisfy both the perimeter condition and the triangle inequality.

Possible Side Lengths

  1. a=1,b=3,c=3a = 1, b = 3, c = 3: This satisfies 1+3+3=71 + 3 + 3 = 7, and the triangle inequalities hold:

    • 1+3>31 + 3 > 3
    • 1+3>31 + 3 > 3
    • 3+3>13 + 3 > 1
  2. a=2,b=2,c=3a = 2, b = 2, c = 3: This satisfies 2+2+3=72 + 2 + 3 = 7, and the triangle inequalities hold:

    • 2+2>32 + 2 > 3
    • 2+3>22 + 3 > 2
    • 2+3>22 + 3 > 2

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

  • (1,3,3)(1, 3, 3)
  • (2,2,3)(2, 2, 3)

Step 3: Calculating the Area Using Heron’s Formula

Heron’s formula for the area of a triangle with sides aa, bb, and cc is: Area=s(sa)(sb)(sc)\text{Area} = \sqrt{s(s-a)(s-b)(s-c)} where s=a+b+c2s = \frac{a + b + c}{2} is the semi-perimeter.

For (1,3,3)(1, 3, 3):

  • Semi-perimeter: s=1+3+32=3.5s = \frac{1 + 3 + 3}{2} = 3.5
  • Area: Area=3.5(3.51)(3.53)(3.53)=3.5×2.5×0.5×0.51.48\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)(2, 2, 3):

  • Semi-perimeter: s=2+2+32=3.5s = \frac{2 + 2 + 3}{2} = 3.5
  • Area: Area=3.5(3.52)(3.52)(3.53)=3.5×1.5×1.5×0.52.90\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)(2, 2, 3) has the maximum possible area of approximately 2.902.90 square units.

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


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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangles
Optimization
Triangle Inequality
Area Calculation

Formulas

Heron's formula for area of a triangle: Area = sqrt(s(s-a)(s-b)(s-c))
Semi-perimeter formula: s = (a + b + c) / 2

Theorems

Triangle Inequality

Suitable Grade Level

Grades 9-10