To find the range of values for the third side of a triangle when two sides are given, we can use the Triangle Inequality Theorem, which states:
- The sum of the lengths of any two sides must be greater than the length of the third side.
- The difference between the lengths of any two sides must be less than the length of the third side.
Let the lengths of the two given sides be \( a = 42.7 \) mm and \( b = 38.03 \) mm. Let \( c \) be the length of the third side. The inequalities from the Triangle Inequality Theorem will be:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Now we can substitute the values of \( a \) and \( b \).
First Inequality:
\[ 42.7 + 38.03 > c \] \[ 80.73 > c \quad \Rightarrow \quad c < 80.73 \]
Second Inequality:
\[ 42.7 + c > 38.03 \] \[ c > 38.03 - 42.7 \] \[ c > -4.67 \quad \Rightarrow \quad c > 0 \quad \text{(since side lengths must be positive)} \]
Third Inequality:
\[ 38.03 + c > 42.7 \] \[ c > 42.7 - 38.03 \] \[ c > 4.67 \]
Combining Inequalities:
From the second and third inequalities, we have: \[ c > 4.67 \quad \text{and} \quad c < 80.73 \]
Thus, the range of values for the third side \( c \) is: \[ 4.67 < c < 80.73 \]
So the final answer is: \[ c \in (4.67, 80.73) \text{ mm} \]