Math Problem Statement
A triangle has two sides of length 8 and 18. 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, we can use the triangle inequality theorem. The 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.
Given the two sides are 8 and 18, let's denote the length of the third side as . We have three inequalities based on the triangle inequality:
- (the sum of the two known sides must be greater than the third side)
- (the sum of one known side and the third side must be greater than the other known side)
- (the sum of the other known side and the third side must be greater than the first known side)
Solving each inequality:
-
From , we get:
-
From , we get:
-
From , we get: But this condition is automatically satisfied since .
Combining the conditions:
The third side must satisfy:
Therefore, the largest whole-number value for is 25.
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Follow-up questions:
- How would the smallest possible third side be calculated using the triangle inequality?
- Can the triangle inequality theorem be applied to non-triangular shapes?
- How can we calculate the area of a triangle given two sides and the included angle?
- What happens to the shape of a triangle when the third side is very small or very large?
- How does the Pythagorean theorem relate to right triangles in the context of side lengths?
Tip:
Always check if a triangle inequality holds before attempting to calculate angles, area, or other properties.
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Math Problem Analysis
Mathematical Concepts
Triangle Inequality
Geometry
Formulas
Triangle Inequality: a + b > c
Theorems
Triangle Inequality Theorem
Suitable Grade Level
Grades 6-8
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