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

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:

1. The sum of the lengths of any two sides must be greater than the length of the third side.
2. 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:

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

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:
\[
c \in (4.67, 80.73) \text{ mm}
\]