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)

Question

Responses

4.67<x<42.3
4.67 less than x less than 42.3

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<80.73

Answer 1

To find the range of values for the third side of a triangle when two sides measure \( a = 42.7 \) mm and \( b = 38.03 \) mm, we can use the triangle inequality theorem. According to this theorem, for any triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the third side), the following inequalities must hold:

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

Let's apply these inequalities.

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

2. From \( a + c > b \): \[ 42.7 + c > 38.03 \implies c > 38.03 - 42.7 \implies c > -4.67 \] Since \( c \) must be a positive value, this does not limit \( c \).

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

Combining the results from the inequalities, we find:

\[ 4.67 < c < 80.73 \]

Therefore, the range of values for the third side of the triangle is:

\( 4.67 < x < 80.73 \)

So the correct response is:

4.67 < x < 80.73