Two sides of a triangle measure 5in and 12in. What could be the length of the third side?

To determine the possible lengths of the third side of a triangle with side lengths of 5 inches and 12 inches, we can use the triangle inequality theorem. This 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.

Let’s denote the lengths of the sides as follows:

• A = 5 inches
• B = 12 inches
• C = length of the third side

According to the triangle inequality, we need to satisfy three conditions:

1. A + B > C
2. A + C > B
3. B + C > A

Plugging in the values we know:

1. 5 + 12 > C ⟹ 17 > C or C < 17
2. 5 + C > 12 ⟹ C > 7
3. 12 + C > 5 ⟹ C > -7 (which is always true for positive lengths)

From the first inequality, we have that C must be less than 17, and from the second inequality we find that C must be greater than 7. Therefore, the possible lengths for the third side must satisfy:

7 < C < 17

In conclusion, the length of the third side can be any value greater than 7 inches but less than 17 inches.