Question

If \( k \) is an integer and \( 2<k<7 \), for how many different values of \( k \) is there a triangle with sides of lengths 2, 7, and \( k \)?

• A. One
• B. Two
• C. Three
• D. Four
• E. Five

Hint:

Use the triangle inequality theorem to find valid side lengths for a triangle.

Answer 1

The Correct Option is C

Step 1: Use the triangle inequality theorem.
For a set of three sides to form a triangle, the sum of the lengths of any two sides must be greater than the third side.
Step 2: Apply the triangle inequality.
Let the three sides of the triangle be 2, 7, and \( k \). The inequalities we need to check are: \[ 2 + 7>k \quad \implies \quad k<9 \] \[ 2 + k>7 \quad \implies \quad k>5 \] \[ 7 + k>2 \quad \implies \quad k>-5 \text{ (which is always true for } k>5) \] Step 3: Conclusion.
Thus, \( k \) must satisfy: \[ 5<k<9 \] The integer values of \( k \) are 6 and 7, so there are 3 possible values for \( k \). \[ \boxed{3} \]