Final answer:
The range of possible lengths for the third side x of the triangle, given the two other side lengths of 34 inches and 67.5 inches, is from more than 33.5 inches to less than 101.5 inches.
Step-by-step explanation:
The range of possible lengths for the third side, x, of a triangle when the lengths of two sides are given can be determined using 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. Conversely, the difference of the lengths of any two sides must be less than the length of the third side.
Given the side lengths of 34 inches and 67.5 inches, the range of the third side x can be calculated as:
- For the lower bound, x must be greater than the difference between the two given side lengths:
- x > 67.5 inches - 34 inches = 33.5 inches
- For the upper bound, x must be less than the sum of the two given side lengths:
- x < 67.5 inches + 34 inches = 101.5 inches
Therefore, the range of possible lengths for the third side x is:
33.5 inches < x < 101.5 inches