Triangle Inequality Theorem and Finding the Range of the Third Side
The Triangle Inequality Theorem is a fundamental concept in geometry that helps determine if a set of three side lengths can form a triangle. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem can be used to find the range of possible lengths for the third side of a triangle when the lengths of two sides are given.
Here is how the Triangle Inequality Theorem can be applied to find the range of the third side:

Step 1: Identify the Given Side Lengths
- Determine the lengths of the two sides that are given in the problem.

Step 2: Apply the Triangle Inequality Theorem
- Add the lengths of the two given sides.
- The sum of the two given sides must be greater than the length of the third side for the triangle to be valid.
- The difference between the sum of the two given sides and the length of the third side gives the range of possible lengths for the third side.

Step 3: Determine the Range of the Third Side
- By applying the Triangle Inequality Theorem, you can find the minimum and maximum possible lengths for the third side.
- The minimum length is found by subtracting the length of the longer given side from the length of the shorter given side.
- The maximum length is found by adding the lengths of the two given sides.
By following these steps and applying the Triangle Inequality Theorem, you can determine the range of possible lengths for the third side of a triangle when two sides are given. This theorem is essential for understanding the relationships between the sides of a triangle and ensuring the triangle's validity.