Question

Which triangle has an unknown side length measuring 7 units?(1 point)
Responses

A 4, √2

B √74, 5

C √43, √6

D 5, √18

Answer 1

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:

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

Let's consider each option and see which one can accommodate a side length of 7 units.

**Option A: Sides are 4 and \(\sqrt{2}\)**

1. \(4 + \sqrt{2} > 7\) (False)
2. \(4 + 7 > \sqrt{2}\) (True)
3. \(\sqrt{2} + 7 > 4\) (True)

Since the first inequality fails, this option cannot form a triangle.

**Option B: Sides are \(\sqrt{74}\) and 5**

1. \(\sqrt{74} + 5 > 7\) (True)
2. \(\sqrt{74} + 7 > 5\) (True)
3. \(5 + 7 > \sqrt{74}\) (True)

All inequalities hold, so this option can form a triangle.

**Option C: Sides are \(\sqrt{43}\) and \(\sqrt{6}\)**

1. \(\sqrt{43} + \sqrt{6} > 7\) (True, since both sides are greater than 7)
2. \(\sqrt{43} + 7 > \sqrt{6}\) (True)
3. \(\sqrt{6} + 7 > \sqrt{43}\) (False)

Since the third inequality fails, this option cannot form a triangle.

**Option D: Sides are 5 and \(\sqrt{18}\)**

1. \(5 + \sqrt{18} > 7\) (True)
2. \(5 + 7 > \sqrt{18}\) (True)
3. \(\sqrt{18} + 7 > 5\) (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!