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

Answer 1

The correct option is A One

In a triangle, the sum of the smaller two sides must be larger than the largest side.

For $k$ values $3,4,5$, and $6$, the only triangle possible is $2,7$, and $k=6$ because only $2+6>7$. For $k$ values $3,4$, and $5$, the sum of the smaller two sides is not larger than the third side; thus, $6$ is the only possible value of $k$ that satisfies the conditions.

The correct answer is A.