The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12, which inequality represents the possible lengths, x, of the third side of the triangle?
Question
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of 6 and 12, which inequality represents the possible lengths, x, of the third side of the triangle?
Solution
The triangle inequality 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. In this case, we have two sides with lengths 6 and 12.
We can create two inequalities based on this theorem:
- The sum of the lengths of the first and second sides must be greater than the length of the third side. This gives us:
6 + 12 > x
Which simplifies to:
18 > x
- The sum of the lengths of the first and third sides must be greater than the length of the second side. This gives us:
6 + x > 12
Which simplifies to:
x > 6
Therefore, the possible lengths, x, of the third side of the triangle must be greater than 6 but less than 18. This can be written as:
6 < x < 18
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