If we call the unknown side x, we have a triangle with sides of 8, 13, and x inches. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. There are two possible scenarios here.
The 13-inch side is the greatest side, which means x is either the shortest side or the middle side.
The 13-inch side is the middle side, which means x must be the greatest side.
The 13-inch Side Is Greatest
If the 13-inch side is the greatest, then by the Triangle Inequality Theorem the sum of x and 8 must be greater than 13. We can write this as the following inequality.
x+8 > 13 ⇔ x > 5
x must be greater than 5 inches.
The x-inch Side Is Greatest
If the x-inch side is the greatest, then according to the Triangle Inequality Theorem the sum of 8 and 13 must be greater than x. We can write this as the following inequality.
8+13 > x ⇔ x < 21
x must be less than 21 inches.
Conclusion
Depending on which side is the longest, the third side must be between 5 and 21 inches.
5 inches < Third side< 21 inches
