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)%0D%0AResponses%0D%0A%0D%0A4.67<x<80.73%0D%0A4.67 less than x less than 80.73%0D%0A%0D%0A4.4<x<80.73%0D%0A4.4 less than x less than 80.73%0D%0A%0D%0A4.67<x<42.3%0D%0A4.67 less than x less than 42.3%0D%0A%0D%0A4.04<x<80.1

Answer 1

To find the range of values for the third side \( x \) of a triangle given two sides measuring 42.7 mm and 38.03 mm, we can use the triangle inequality theorem. This 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 of the triangle be:
- \( a = 42.7 \) mm
- \( b = 38.03 \) mm
- \( c = x \) (the unknown side)

We can set up the following inequalities based on the triangle inequality theorem:

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

2. \( a + c > b \)
\[
42.7 + x > 38.03
\]
\[
x > 38.03 - 42.7
\]
\[
x > -4.67 \quad \text{(since the side length must be positive, we drop this)}
\]

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

Combining these inequalities, we find that the third side \( x \) must satisfy:
\[
4.67 < x < 80.73
\]

Thus, the correct response is:
**4.67 < x < 80.73** or **4.67 less than x less than 80.73**.