Question
Find the length of the third side. If necessary, round to the nearest tenth.
13
5
13
5
Answers
GPT-4o mini
To determine the length of the third side of a triangle when two sides are given, we can use the properties of triangles and 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, and conversely, the length of any one side must be less than the sum of the lengths of the other two sides.
Let’s denote the lengths of the given sides as \(a = 13\) and \(b = 5\), and the length of the third side as \(c\).
According to the triangle inequality theorem:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Using these inequalities, we can find the possible range for \(c\):
1. From \(13 + 5 > c\), we have:
\[
c < 18
\]
2. From \(13 + c > 5\), we have:
\[
c > 5 - 13 \implies c > -8 \quad \text{(This condition is always true as \(c\) must be positive)}
\]
3. From \(5 + c > 13\), we have:
\[
c > 13 - 5 \implies c > 8
\]
Combining these results, we find:
\[
8 < c < 18
\]
Therefore, the length of the third side \(c\) can be any value between 8 and 18. If you need a specific length, you can choose a value within that range based on additional constraints or context given in a problem. If you simply need a single value, you could use the midpoint, which would be \(c = 13\), but any value from this range would satisfy the triangle's side length requirements.
Let’s denote the lengths of the given sides as \(a = 13\) and \(b = 5\), and the length of the third side as \(c\).
According to the triangle inequality theorem:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Using these inequalities, we can find the possible range for \(c\):
1. From \(13 + 5 > c\), we have:
\[
c < 18
\]
2. From \(13 + c > 5\), we have:
\[
c > 5 - 13 \implies c > -8 \quad \text{(This condition is always true as \(c\) must be positive)}
\]
3. From \(5 + c > 13\), we have:
\[
c > 13 - 5 \implies c > 8
\]
Combining these results, we find:
\[
8 < c < 18
\]
Therefore, the length of the third side \(c\) can be any value between 8 and 18. If you need a specific length, you can choose a value within that range based on additional constraints or context given in a problem. If you simply need a single value, you could use the midpoint, which would be \(c = 13\), but any value from this range would satisfy the triangle's side length requirements.