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<42.3%0D%0A4.67%0D%0A<%0D%0A𝑥%0D%0A<%0D%0A42.3%0D%0A4.67 less than x less than 42.3%0D%0A%0D%0A4.4<x<80.73%0D%0A4.4%0D%0A<%0D%0A𝑥%0D%0A<%0D%0A80.73%0D%0A4.4 less than x less than 80.73%0D%0A%0D%0A4.04<x<80.1%0D%0A4.04%0D%0A<%0D%0A𝑥%0D%0A<%0D%0A80.1%0D%0A4.04 less than x less than 80.1%0D%0A%0D%0A4.67<x<80.73

Answer 1

To determine the range of possible 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. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This gives us three inequalities to consider:

1. \( x + 38.03 > 42.7 \)
2. \( x + 42.7 > 38.03 \)
3. \( 42.7 + 38.03 > x \)

Let's solve each inequality:

1. \( x + 38.03 > 42.7 \)
\[
x > 42.7 - 38.03
\]
\[
x > 4.67
\]

2. \( x + 42.7 > 38.03 \)
\[
x > 38.03 - 42.7
\]
Since this simplifies to \( x > -4.67 \), it's always true for positive lengths and can be ignored.

3. \( 42.7 + 38.03 > x \)
\[
80.73 > x
\]
or
\[
x < 80.73
\]

Combining these results, we get:

\[
4.67 < x < 80.73
\]

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