To find the range of possible values for the third side of a triangle, you 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.
Given the sides \( a = 7 \), \( b = 11 \), and \( c \) (the third side), the triangle inequalities can be written as follows:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's apply these inequalities:
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From \( a + b > c \): \[ 7 + 11 > c \implies 18 > c \implies c < 18 \]
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From \( a + c > b \): \[ 7 + c > 11 \implies c > 11 - 7 \implies c > 4 \]
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From \( b + c > a \): \[ 11 + c > 7 \implies c > 7 - 11 \implies c > -4 \quad (\text{This condition is always true since } c > 4) \]
Combining the two relevant inequalities, we find: \[ 4 < c < 18 \]
Thus, the range of values for the third side \( c \) of the triangle is: \[ (4, 18) \]