Math Problem Statement
A triangle has two sides of length 8 and 12. What is the largest possible whole-number length for the third side?
Solution
To find the largest possible whole-number length for the third side of a triangle with two sides measuring 8 and 12, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let the third side be denoted by . The triangle inequality gives us three conditions:
- →
- →
- → (which is always true since is positive)
Thus, must satisfy:
The largest possible whole-number length for the third side is .
Would you like more details or have any questions?
Here are five related questions to expand on this topic:
- What is the smallest possible value for the third side of the triangle?
- How does the triangle inequality theorem apply to different types of triangles?
- If the triangle were a right triangle, how would the third side change?
- How would the triangle’s area be affected by changing the third side?
- What happens to the angles in a triangle as you change the length of one side?
Tip: Always remember that the sum of any two sides of a triangle must be greater than the third side to form a valid triangle.
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Math Problem Analysis
Mathematical Concepts
Geometry
Triangle Inequality Theorem
Formulas
a + b > c
a + c > b
b + c > a
Theorems
Triangle Inequality Theorem
Suitable Grade Level
Grades 7-9
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