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)
Responses

4.67<x<42.3

4.04<x<80.1

4.4<x<80.73

4.67<x<80.73

Answer 1

To find the range of values for the third side of a triangle given two sides, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let \( a = 42.7 \) mm and \( b = 38.03 \) mm, and let \( c \) be the length of the third side. The inequalities we need to satisfy are:

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

Let's calculate each of these inequalities:

1. \( a + b > c \)
\[
42.7 + 38.03 > c \implies 80.73 > c \implies c < 80.73
\]

2. \( a + c > b \)
\[
42.7 + c > 38.03 \implies c > 38.03 - 42.7 \implies c > -4.67
\]

Since side lengths must be positive, we can ignore anything less than zero, so we focus on:

3. \( b + c > a \)
\[
38.03 + c > 42.7 \implies c > 42.7 - 38.03 \implies c > 4.67
\]

Combining these inequalities, we have:

\[
4.67 < c < 80.73
\]

Thus, the range of values for the third side \( c \) is:

\[
4.67 < c < 80.73
\]

From the provided options, the correct choice is:

**4.67 < x < 80.73**.