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<80.73 4.67 less than x less than 80.73 4.4<x<80.73 4.4 less than x less than 80.73 4.04<x<80.1 4.04 less than x less than 80.1 4.67<x<42.3

Answer 1

To determine the range of values for the third side \( x \) of a triangle when the other two sides are given as 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.

We denote the two sides as:

• \( a = 42.7 \) mm
• \( b = 38.03 \) mm
• \( c = x \) (the third side we need to find)

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

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

Now let's apply these inequalities:

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

2. From \( a + c > b \): \[ 42.7 + x > 38.03 \ x > 38.03 - 42.7 \ x > -4.67 \ \text{Since } x \text{ must be positive, we ignore this condition.} \]

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

Combining the inequalities, we have: \[ 4.67 < x < 80.73 \]

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

4.67 < x < 80.73.