To determine which triangle has an unknown side length measuring 7 units, we can apply the triangle inequality theorem. The triangle inequality states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following must hold true:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's consider each option and see which one can accommodate a side length of 7 units.
Option A: Sides are 4 and \(\sqrt{2}\)
- \(4 + \sqrt{2} > 7\) (False)
- \(4 + 7 > \sqrt{2}\) (True)
- \(\sqrt{2} + 7 > 4\) (True)
Since the first inequality fails, this option cannot form a triangle.
Option B: Sides are \(\sqrt{74}\) and 5
- \(\sqrt{74} + 5 > 7\) (True)
- \(\sqrt{74} + 7 > 5\) (True)
- \(5 + 7 > \sqrt{74}\) (True)
All inequalities hold, so this option can form a triangle.
Option C: Sides are \(\sqrt{43}\) and \(\sqrt{6}\)
- \(\sqrt{43} + \sqrt{6} > 7\) (True, since both sides are greater than 7)
- \(\sqrt{43} + 7 > \sqrt{6}\) (True)
- \(\sqrt{6} + 7 > \sqrt{43}\) (False)
Since the third inequality fails, this option cannot form a triangle.
Option D: Sides are 5 and \(\sqrt{18}\)
- \(5 + \sqrt{18} > 7\) (True)
- \(5 + 7 > \sqrt{18}\) (True)
- \(\sqrt{18} + 7 > 5\) (True)
All inequalities hold, so this option can also form a triangle.
By analyzing the options, we see that both options B and D can accommodate a side length of 7 units. However, since the question asks for the triangle with an unknown side length measuring 7, the answer is:
B: \(\sqrt{74}, 5\)
Should you have any other questions or need further clarity, feel free to ask!
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