To find the range of values for the third side of a triangle when given two sides \( a \) and \( b \), we can use the triangle inequality theorem. The theorem states:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Where \( c \) is the length of the third side. Let's denote the two given sides as follows:
- \( a = 42.7 , \text{mm} \)
- \( b = 38.03 , \text{mm} \)
Using the triangle inequalities:
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From \( a + b > c \): \[ 42.7 + 38.03 > c \Rightarrow 80.73 > c \Rightarrow c < 80.73 \]
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From \( a + c > b \): \[ 42.7 + c > 38.03 \Rightarrow c > 38.03 - 42.7 \Rightarrow c > -4.67 \] Since the length of a side cannot be negative, we have \( c > 0 \).
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From \( b + c > a \): \[ 38.03 + c > 42.7 \Rightarrow c > 42.7 - 38.03 \Rightarrow c > 4.67 \]
Combining these inequalities, we find: \[ 4.67 < c < 80.73 \]
Thus, the range of values for the third side \( c \) is: \[ 4.67 < c < 80.73 \]
Therefore, the correct response is: 4.67 < x < 80.73.