To determine the possible range of values for the third side of a triangle when the other two sides are known, we use the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Given:
- Side [tex]\(a = 42.7\)[/tex] mm
- Side [tex]\(b = 38.03\)[/tex] mm
We need to find the range of possible values for the third side [tex]\(c\)[/tex].
### Step-by-Step Solution:
1. Calculate the lower bound:
[tex]\[
c > |a - b|
\][/tex]
Substituting the values:
[tex]\[
c > |42.7 - 38.03|
\][/tex]
[tex]\[
c > 4.67
\][/tex]
Therefore, the lower bound for [tex]\(c\)[/tex] is 4.67 mm.
2. Calculate the upper bound:
[tex]\[
c < a + b
\][/tex]
Substituting the values:
[tex]\[
c < 42.7 + 38.03
\][/tex]
[tex]\[
c < 80.73
\][/tex]
Therefore, the upper bound for [tex]\(c\)[/tex] is 80.73 mm.
Combining these inequalities, we find that the third side [tex]\(c\)[/tex] must satisfy:
[tex]\[
4.67 < c < 80.73
\][/tex]
Thus, the range of values for the third side of the triangle is:
[tex]\[
4.67 < c < 80.73
\][/tex]
Among the given options, the correct one encapsulating this range is:
[tex]\[
\boxed{4.67 < x < 80.73}
\][/tex]
This is the correct range for the possible values of the third side of the triangle.
