Two sides of a triangle measure 34 and 51. Complete the inequality that indicates the possible values of the third side of the triangle using the Triangle Inequality Theorem. (1 point)

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Answer 1

To determine the possible values of the third side \( c \) of a triangle with the other two sides measuring \( a = 34 \) and \( b = 51 \), we can use the Triangle Inequality Theorem, which states that for any triangle with sides \( a \), \( b \), and \( c \):

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

We can apply these inequalities:

1. From \( a + b > c \): \[ 34 + 51 > c \implies 85 > c \implies c < 85 \]

2. From \( a + c > b \): \[ 34 + c > 51 \implies c > 51 - 34 \implies c > 17 \]

3. From \( b + c > a \): \[ 51 + c > 34 \implies c > 34 - 51 \implies c > -17 \] (This last inequality does not provide additional constraints since \( c > 17 \) is a stronger condition.)

Putting these inequalities together, we have the final range for the third side \( c \):

\[ 17 < c < 85 \]

Thus, the inequality indicating the possible values of the third side \( c \) of the triangle is:

\[ 17 < c < 85 \]