The lengths 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
The answer is d. 1<n<25
Thank you R scott help me alot
Well, well, well, looks like we've got ourselves a triangle problem! Let me put on my clown wig and answer this with a little bit of humor.
Now, let's think about this. The longest side in a triangle is always shorter than the sum of the lengths of the other two sides. So, in this case, we have 12 and 13 as the lengths of the other two sides.
If n is less than 12, then the sum of 12 and 13 would be greater than n. So, n cannot be less than 12.
If n is equal to 12, then the sum of 12 and 13 is 25. This is okay because the longest side can be equal to the sum of the other two sides.
If n is greater than 12, then the sum of 12 and 13 would be less than n. So, n can definitely be greater than 12.
Based on this, we can say that 1 < n < 25. So, the correct answer is option D. 1 < n < 25.
To determine the conditions for the length of the unknown side of the triangle, we can use 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.
So, for the sides of lengths 12, 13, and n:
12 + 13 > n
25 > n
Therefore, we know that n must be less than 25, since if it were equal to or greater than 25, the triangle inequality theorem would not hold.
Now, let's examine the provided options to determine which is true:
A. n ≥ 1 - Since n must be less than 25, we cannot conclude that it is greater than or equal to 1. So, this option is not necessarily true.
B. n < 13 - From the triangle inequality theorem, we know that n must be less than 25. This means it is definitely less than 13, so this option is true.
C. 1 < n < 13 - This option includes the range of values for n that satisfy the triangle inequality theorem, so it is true.
D. 1 < n < 25 - This option includes the entire possible range of values for n, so it is also true.
Therefore, the correct options are B, C, and D.
To determine which statement must be true, we need to analyze the given information and the properties of triangles.
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's consider the given sides: 12, 13, and n.
For the sum of the lengths of any two sides to be greater than the length of the third side:
1. 12 + 13 > n
2. 12 + n > 13
3. 13 + n > 12
Simplifying these inequalities, we get:
1. n > -1
2. n > 1
3. n > -1
So, the common condition from the three inequalities is that n must be greater than 1.
Therefore, the correct answer is B. N<13.
Explanation: To answer this question, we must apply 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 remaining side. By setting up and solving the relevant inequalities, we can determine the valid range of values for n.