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 , , and with: and each side length , , must be a positive integer. Additionally, the side lengths must satisfy the triangle inequality, which states:
Step 2: Testing Possible Combinations of Integer Side Lengths
Since , let's try values for , , and that satisfy both the perimeter condition and the triangle inequality.
Possible Side Lengths
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: This satisfies , and the triangle inequalities hold:
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: This satisfies , and the triangle inequalities hold:
Other combinations, like , do not satisfy the triangle inequality, so the valid triangles are:
Step 3: Calculating the Area Using Heron’s Formula
Heron’s formula for the area of a triangle with sides , , and is: where is the semi-perimeter.
For :
- Semi-perimeter:
- Area:
For :
- Semi-perimeter:
- Area:
Conclusion
The triangle with sides has the maximum possible area of approximately square units.
Would you like further clarification or a deeper explanation on any part of this solution?
Here are some related questions:
- How does Heron’s formula work, and why is it useful for finding triangle areas?
- What are other methods for finding the area of a triangle?
- How do the triangle inequality conditions ensure valid triangles?
- Could we solve this problem using coordinate geometry instead?
- 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
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