To determine the range of values for the third side of a triangle when the other two sides are given, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side.
Given:
- Side \( a = 7 \)
- Side \( b = 11 \)
- Side \( c \) (the third side)
The triangle inequality provides the following inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's apply these inequalities:
-
From \( a + b > c \): \[ 7 + 11 > c \implies 18 > c \implies c < 18 \]
-
From \( a + c > b \): \[ 7 + c > 11 \implies c > 11 - 7 \implies c > 4 \]
-
From \( b + c > a \): \[ 11 + c > 7 \implies c > 7 - 11 \implies c > -4 \] (This inequality does not provide a useful lower bound since \( c > 4 \) is already stricter.)
Combining the useful inequalities, we find that: \[ 4 < c < 18 \]
Thus, the range of values for the third side \( c \) is: \[ 4 < c < 18 \]
So the final answer is: \[ 4 < n < 18 \]