Question

A) 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 < x < 80.73
4.67 < x < 80.73
4.67 < x < 42.3
4.04 < x < 80.1

Answer 1

To determine the range of values for the third side of a triangle when two sides are known, 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.

Let the sides of the triangle be:
- \( a = 42.7 \) mm
- \( b = 38.03 \) mm
- \( c = x \) (the length of the third side)

The triangle inequalities are:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Now, let's apply these inequalities:

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

2. From \( a + c > b \):
\[
42.7 + x > 38.03 \implies x > 38.03 - 42.7 \implies x > -4.67
\]
This doesn't provide a meaningful lower bound since \( x \) is positive.

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

Combining the two meaningful inequalities from above:
- \( x > 4.67 \)
- \( x < 80.73 \)

Thus, the range of values for the third side \( x \) is:
\[
4.67 < x < 80.73
\]

The correct answer is:
**4.67 < x < 80.73**.