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)

Answer 1

The Triangle Inequality 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 two sides of the triangle measuring 34 and 51, let's denote the length of the third side as \( c \).

According to the theorem, we can set up the following inequalities:

1. \( 34 + 51 > c \)
2. \( 34 + c > 51 \)
3. \( 51 + c > 34 \)

Now we can simplify these inequalities:

1. \( 85 > c \) or \( c < 85 \)
2. \( c > 51 - 34 \) or \( c > 17 \)
3. \( c > 34 - 51 \) or \( c > -17 \) (this inequality is always true since \( c \) must be positive).

The relevant inequalities for the third side \( c \) are \( c < 85 \) and \( c > 17 \).

Combining these, the inequality that indicates the possible values of the third side \( c \) is:

\[ 17 < c < 85 \]

Answer 2

Use the image to answer the question.
Find the range of values for the third side of the triangle, x, based on the diagram.
(1 point)
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