Final answer:
The third side x of the triangle must be greater than 15 units and less than 35 units according to the Triangle Inequality Theorem. The range is determined by the sum and difference of the lengths of the other two sides which are 10 and 25 units.
Step-by-step explanation:
To determine the range of possible lengths for the third side x of a triangle when the lengths of the other two sides are given as 10 and 25 units, we apply the Triangle Inequality Theorem. This theorem states that for any triangle, the length of any side must be less than the sum and greater than the difference of the lengths of the other two sides. Therefore, the minimum length of the third side is the difference of the two lengths (25 - 10 = 15 units), and the maximum length is the sum of the two lengths (25 + 10 = 35 units).
The range of the third side, x, is therefore > 15 and < 35 units. These are the values that x must fall between to form a valid triangle.
Step-by-Step Explanation:
- Use the Triangle Inequality Theorem: the sum of the lengths of any two sides must be greater than the length of the remaining side.
- Calculate the minimum length for the third side, x: (25 - 10) = 15 units.
- Calculate the maximum length for the third side, x: (25 + 10) = 35 units.
- The range of lengths for the third side x is from > 15 to < 35 units, meaning x must be greater than 15 units but less than 35 units.