Question

the range of the third side lengths depends on the sun and difference of the two given sides, Refer to the section covering the slides of a triangle for further assistance.

Answer 1

To find the range of possible lengths for the third side of a triangle given two sides, you can use the triangle inequality theorem. This theorem states that, for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following must hold true:

1. \( a + b > c \) (the sum of the lengths of any two sides must be greater than the length of the third side)
2. \( a + c > b \)
3. \( b + c > a \)

Given two sides, let's denote them as \( a \) and \( b \). You want to find the possible lengths for the third side, which we will call \( c \).

From the triangle inequality, we derive:

1. From \( a + b > c \), we find \( c < a + b \).
2. From \( a + c > b \), we rearrange this to get \( c > b - a \) if \( b > a \) (or \( c > a - b \) if \( a > b \)).
3. From \( b + c > a \), we rearrange this to \( c > a - b \) if \( a > b \) (or \( c > b - a \) if \( b > a \)).

Combining these inequalities, we get two cases for the range of \( c \):

1. If \( a \geq b \):
- The valid range for \( c \) is:
\[
c > a - b \quad \text{and} \quad c < a + b
\]
So, the range is:
\[
a - b < c < a + b
\]

2. If \( b > a \):
- The valid range for \( c \) would be the same, but written considering \( b \) as the larger side:
\[
b - a < c < a + b
\]

In conclusion, irrespective of which side is longer, the range for the length of the third side \( c \) is:
\[
|a - b| < c < a + b
\]
This means the length of the third side must be greater than the absolute difference between the two given sides and less than the sum of the two sides.