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)

4.4 80.73

4.04 80.1

4.67 80.73

4.67<< 42.3

Answer 1

To find the range of values for the third side of a triangle given two sides, you 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 sides be:
- \( a = 42.7 \, \text{mm} \)
- \( b = 38.03 \, \text{mm} \)
- \( c \) = the length of the third side.

According to the triangle inequality theorem, we have the following three conditions:

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

Now applying these to our sides:

1. From the first condition:
\[
42.7 + 38.03 > c \implies 80.73 > c \implies c < 80.73
\]

2. From the second condition:
\[
42.7 + c > 38.03 \implies c > 38.03 - 42.7 \implies c > -4.67 \text{ (which is always true, since c must be positive)}
\]

3. From the third condition:
\[
38.03 + c > 42.7 \implies c > 42.7 - 38.03 \implies c > 4.67
\]

Combining these results gives us the range for \( c \):
\[
4.67 < c < 80.73
\]

Thus, the range of possible values for the third side is \( (4.67, 80.73) \).

Therefore, the answer you are looking for is:
**4.67 80.73**.