Answer:
THE ANSWER IS GIVEN BELOW
Step-by-step explanation:
To determine the range of possible lengths for the third side of the triangle, we can 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 third side.
Given that the triangle has sides of lengths 4 and 13, let's denote the length of the third side as \(x\).
According to the triangle inequality theorem:
1. The sum of the lengths of the two smaller sides (4 and \(x\)) must be greater than the length of the largest side (13):
\[4 + x > 13\]
2. Similarly, the sum of the lengths of the two smaller sides (13 and \(x\)) must be greater than the length of the largest side (4):
\[13 + x > 4\]
Now, let's solve these inequalities for \(x\):
1. \(4 + x > 13\)
Subtract 4 from both sides:
\[x > 13 - 4\]
\[x > 9\]
2. \(13 + x > 4\)
Subtract 13 from both sides:
\[x > 4 - 13\]
\[x > -9\]
Therefore, the third side must be greater than 9 and less than 13.
So, the range for the length of the third side is \(9 < x < 13\).