To find the range of values for the third side of a triangle when two sides are known, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let the lengths of two sides be \( a = 42.7 \) mm and \( b = 38.03 \) mm, and let the length of the third side be \( c \).
According to the triangle inequality, we have:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Now we can apply these inequalities to find the range for \( c \):
-
From \( a + b > c \): \[ 42.7 + 38.03 > c \] \[ 80.73 > c \quad \text{or} \quad c < 80.73 \]
-
From \( a + c > b \): \[ 42.7 + c > 38.03 \] \[ c > 38.03 - 42.7 \] \[ c > -4.67 \quad \text{(which is always true since lengths are positive)} \]
-
From \( b + c > a \): \[ 38.03 + c > 42.7 \] \[ c > 42.7 - 38.03 \] \[ c > 4.67 \]
Combining these results, we find that the range of values for the third side \( c \) is: \[ 4.67 < c < 80.73 \]
Thus, the range of values for the third side of the triangle is \( (4.67 \text{ mm}, 80.73 \text{ mm}) \).