The lengths of the sides of a triangle are 12, 13, and n. Which of the following must be true?
(1 point)
Responses
n ≥ 1
n ≥ 1
n < 13
n < 13
1 < n < 13
1 < n < 13
1 < n < 25
Using the triangle inequality theorem, 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 sides of lengths 12, 13, and n.
To satisfy the triangle inequality theorem, we need:
12 + 13 > n
25 > n
Therefore, we can conclude that 1 < n < 25.
Therefore, the statement "1 < n < 25" must be true.
To determine which of the statements must be true, we need to consider the triangle inequality theorem, which 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 the side lengths 12, 13, and n. Based on the triangle inequality theorem:
12 + 13 > n
25 > n
From this, we can conclude that:
1. n < 25
However, we cannot determine the exact range of values for n based on this information alone. Therefore, it is not correct to say that 1 < n < 13 or n < 13 must be true.
In summary, the only statement that must be true is:
n < 25
To determine which of the given statements must be true, you need to use the property of triangle inequalities. In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's analyze each statement:
1. n ≥ 1: This statement must be true because in any triangle, the length of a side cannot be negative or zero.
2. n < 13: This statement is not necessarily true. While n could possibly be less than 13, it is not a requirement in this case.
3. 1 < n < 13: This statement is also not necessarily true. While n could possibly fall within this range, it is not a requirement in this case.
4. 1 < n < 25: This statement is also not necessarily true for the same reason mentioned above. Although n could fall within this range, it is not a requirement in this case.
Therefore, the only statement that must be true is n ≥ 1.