Question

Find the range of values for the third side of a triangle if two of its sides measure 42.7 mm and 38.03 mm.

Answer 1

In any triangle, the lengths of the sides must satisfy the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following must hold true:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Let us denote the two given sides as \(a = 42.7 \, \text{mm}\) and \(b = 38.03 \, \text{mm}\). We will denote the third side as \(c\).

We need to apply the triangle inequalities:

1. For \(a + b > c\):
\[
42.7 + 38.03 > c \quad \Rightarrow \quad 80.73 > c \quad \Rightarrow \quad c < 80.73
\]

2. For \(a + c > b\):
\[
42.7 + c > 38.03 \quad \Rightarrow \quad c > 38.03 - 42.7 \quad \Rightarrow \quad c > -4.67
\]
Since a side length cannot be negative, this condition does not impose a restriction; we will focus on the next condition.

3. For \(b + c > a\):
\[
38.03 + c > 42.7 \quad \Rightarrow \quad c > 42.7 - 38.03 \quad \Rightarrow \quad c > 4.67
\]

Combining the results from the inequalities, we find:

- The third side \(c\) must satisfy \(c > 4.67\) and \(c < 80.73\).

Thus, the range of values for the third side \(c\) is:
\[
\boxed{(4.67, 80.73)}
\]