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.(1 point)

Answer 1

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:

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

Now we can apply these inequalities to find the range for \( c \):

1. From \( a + b > c \):
\[
42.7 + 38.03 > c
\]
\[
80.73 > c \quad \text{or} \quad c < 80.73
\]

2. From \( a + c > b \):
\[
42.7 + c > 38.03
\]
\[
c > 38.03 - 42.7
\]
\[
c > -4.67 \quad \text{(which is always true since lengths are positive)}
\]

3. 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 \( (4.67 \text{ mm}, 80.73 \text{ mm}) \).