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Question

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)

Check answer Remaining Attempts: 3

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, and vice versa.

Let the lengths of the two known sides be \( a = 34 \) and \( b = 51 \), and let \( c \) be the length of the third side. According to the Triangle Inequality Theorem:

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

Now, applying the values:

1. \( 34 + 51 > c \) → \( 85 > c \) → \( c < 85 \)
2. \( 34 + c > 51 \) → \( c > 51 - 34 \) → \( c > 17 \)
3. \( 51 + c > 34 \) → \( c > 34 - 51 \) → This is always true since \( 34 - 51 \) is negative.

Combining the inequalities from steps 1 and 2, we find:

\[ 17 < c < 85 \]

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

17 < c < 85