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:

A shortcut for the Triangle Inequality Theorem is that the third side `c` must be between the difference and the sum of the other two sides: \(|a-b|<c<a+b\). For sides 2 and 7, we have \(|7-2|<k<7+2\), which simplifies to \(5<k<9\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires the application of the Triangle Inequality Theorem, which defines the relationship between the lengths of the sides of a triangle.
Step 2: Key Formula or Approach:
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. For sides a, b, and c, the following three conditions must be met:

\(a + b>c\)
\(a + c>b\)
\(b + c>a\)
Step 3: Detailed Explanation:
First, identify the possible integer values for k. The condition \(2<k<7\) means k can be 3, 4, 5, or 6.
Now, let's apply the Triangle Inequality Theorem with sides \(a=2\), \(b=7\), and \(c=k\).

\(2 + 7>k \implies 9>k\)
\(2 + k>7 \implies k>5\)
\(7 + k>2 \implies k>-5\) (This is always true for positive k)
We must satisfy both \(k<9\) and \(k>5\). So, the valid range for k is \(5<k<9\).
Now we check which of the possible integer values for k (\(k \in \{3, 4, 5, 6\}\)) fall into this valid range.

Is \(k=3\) in the range \(5<k<9\)? No.
Is \(k=4\) in the range \(5<k<9\)? No.
Is \(k=5\) in the range \(5<k<9\)? No (it's not strictly greater).
Is \(k=6\) in the range \(5<k<9\)? Yes.
Only one value, \(k=6\), satisfies the conditions.
Step 4: Final Answer:
There is only one possible integer value for k.