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:
- A + B > C
- A + C > B
- B + C > A
Plugging in the values we know:
- 5 + 12 > C ⟹ 17 > C or C < 17
- 5 + C > 12 ⟹ C > 7
- 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.