Final answer:
The range for the measure of the third side X in a triangle with the other two sides measuring 5 cm and 8 cm is from greater than 3 cm to less than 13 cm, due to the triangle inequality theorem.
Step-by-step explanation:
To find the range of the possible measure of the third side (X) of a triangle when the other two sides of the triangle measure 5 cm and 8 cm, you 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 remaining side.
Using the triangle inequality theorem:
- The sum of the two shorter sides (5 cm and X) must be greater than the third side (8 cm): 5 + X > 8.
- The sum of the longer side (8 cm) and the third side (X) must be greater than the other shorter side (5 cm): 8 + X > 5.
- The difference between the longer side (8 cm) and the shorter side (5 cm) must be less than the third side (X): X > 8 - 5, which simplifies to X > 3.
From these inequalities, you can see that:
- X must be greater than 3 cm because of the third inequality (X > 3).
- X must be less than 13 cm because of the first inequality rearranged (5 + X > 8 leads to X < 13).
Therefore, the range of X is from greater than 3 cm but less than 13 cm, or 3 cm < X < 13 cm.