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Which triangle has an unknown side length of \/37 units?

4, \/2

\/43, \/ 6

\/74, 5

5, \/18

3 answers

To determine which triangle has an unknown side length of \(\sqrt{37}\) units, we can apply the triangle inequality theorem. The theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following inequalities must hold:

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

We'll evaluate each option to check if \(\sqrt{37}\) can be an unknown side while satisfying the triangle inequality.

  1. Option 1: Sides are \(4\), \(\sqrt{2}\), and \(\sqrt{37}\)

    • \(4 + \sqrt{2} > \sqrt{37}\)
    • \(4 + \sqrt{37} > \sqrt{2}\)
    • \(\sqrt{2} + \sqrt{37} > 4\)

    We can check these inequalities:

    • Approximate values: \(4 + 1.41 \approx 5.41\) (which is greater than \(6.08\) as \(\sqrt{37} \approx 6.08\)), so this inequality fails.

Thus, this triangle cannot have \(\sqrt{37}\) as one of its sides.

  1. Option 2: Sides are \(\sqrt{43}\), \(6\), and \(\sqrt{37}\)

    • \(\sqrt{43} + 6 > \sqrt{37}\)
    • \(\sqrt{43} + \sqrt{37} > 6\)
    • \(6 + \sqrt{37} > \sqrt{43}\)

    Approximate values:

    • \(\sqrt{43} \approx 6.56\) and \(\sqrt{37} \approx 6.08\)
    • \(\sqrt{43} + 6 \approx 12.56\), which is greater than \(6.08\)
    • \(\sqrt{43} + \sqrt{37} \approx 12.64\), which is greater than \(6\)
    • \(6 + \sqrt{37} \approx 12.08\), which is greater than \(\sqrt{43}\)

    All inequalities hold, therefore this triangle is valid.

  2. Option 3: Sides are \(\sqrt{74}\), \(5\), and \(\sqrt{37}\)

    • \(\sqrt{74} + 5 > \sqrt{37}\)
    • \(\sqrt{74} + \sqrt{37} > 5\)
    • \(5 + \sqrt{37} > \sqrt{74}\)

    Approximate values:

    • \(\sqrt{74} \approx 8.60\)
    • \(\sqrt{74} + 5 \approx 13.60\), which is greater than \(6.08\)
    • \(\sqrt{74} + \sqrt{37} \approx 14.68\), which is greater than \(5\)
    • \(5 + \sqrt{37} \approx 11.08\), which is less than \(\sqrt{74}\)

    The last inequality fails, so this triangle cannot have \(\sqrt{37}\) as one of its sides.

  3. Option 4: Sides are \(5\), \(\sqrt{18}\), and \(\sqrt{37}\)

    • \(5 + \sqrt{18} > \sqrt{37}\)
    • \(5 + \sqrt{37} > \sqrt{18}\)
    • \(\sqrt{18} + \sqrt{37} > 5\)

    Approximate values:

    • \(\sqrt{18} \approx 4.24\)
    • \(5 + 4.24 \approx 9.24\), which is greater than \(6.08\)
    • \(5 + 6.08 \approx 11.08\), which is greater than \(4.24\)
    • \(4.24 + 6.08 \approx 10.32\), which is greater than \(5\)

    All inequalities hold, thus this triangle is valid.

Therefore, the triangles that can have an unknown side length of \(\sqrt{37}\) are Option 2 and Option 4.

u have to chose 1 answer

Apologies for the confusion. Based on the given options and the triangle inequality, the triangle that can have an unknown side length of \(\sqrt{37}\) is:

Option 2: \(\sqrt{43}\), \(6\), and \(\sqrt{37}\).

This triangle satisfies all the inequalities required by the triangle inequality theorem.