the length of the sides of a triangle are 12, 13, and n. which of the following must be true?
A. N≥1
B. n<13
C. 1<n<13
D. 1<n<25
To determine the possibilities for the length of the third side of the triangle, we can use 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 sides of length 12, 13, and n.
Using the triangle inequality theorem:
12 + 13 > n (side 1 + side 2 > side 3)
25 > n
So, we know that n must be less than 25.
Therefore, the correct answer is:
D. 1 < n < 25
To determine which of the following statements must be true, we can apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's analyze the given options:
A. N≥1: This statement does not provide any specific information about the triangle's sides. It could potentially be true, but it is not necessarily true in all cases.
B. n<13: This statement suggests that the length of the third side (n) is less than 13. However, since we already know that one side is 13, this statement cannot be true in all cases.
C. 1<n<13: This statement states that the length of the third side (n) falls between 1 and 13. Since the smallest side length is known to be 12, and the sum of the two smaller side lengths (12 + n) must be greater than the largest side length (13), this statement is true.
D. 1<n<25: This statement suggests that the length of the third side (n) falls between 1 and 25. Since the largest side length is known to be 13, and the sum of the two smaller side lengths (12 + n) must be greater than the largest side length (13), this statement is also true.
Therefore, the correct answer is C and D.
To determine which statement must be true, we need to apply the Triangle Inequality Theorem. According to the theorem, for any triangle with side lengths a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.
In this case, we have side lengths of 12, 13, and n. So, we can check the three possible combinations:
1. 12 + 13 > n
2. 12 + n > 13
3. 13 + n > 12
Simplifying these inequalities, we have:
1. 25 > n
2. n > 1
3. n > -1
From these inequalities, we can conclude that n must be greater than 1, meaning n ≥ 2. Therefore, the correct answer is A. N≥1.