Question

select all the correct answers.

which three pairs of measurements are possible side lengths for the triangle?

\\(ab = 4\\), \\(bc = 4\sqrt{3}\\)
\\(bc = 2\sqrt{3}\\), \\(ac = 2\\)
\\(ab = 3\\), \\(ac = 3\sqrt{3}\\)
\\(bc = 10\\), \\(ac = 4\sqrt{3}\\)
\\(ab = 7\\), \\(ac = 14\\)
\\(ab = 11\\), \\(bc = 11\sqrt{3}\\)

Explanation:

Identify the properties of the given triangle

The given triangle \(ABC\) is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) right triangle, where:

• \(AB\) is the shorter leg (opposite the \(30^\circ\) angle).
• \(BC\) is the longer leg (opposite the \(60^\circ\) angle).
• \(AC\) is the hypotenuse (opposite the \(90^\circ\) angle).

The ratio of the side lengths is:

$$ AB : BC : AC = x : x\sqrt{3} : 2x $$

Evaluate each option

• Option 1: \(AB = 4, BC = 4\sqrt{3}\)

If \(AB = x = 4\), then \(BC = x\sqrt{3} = 4\sqrt{3}\). This is possible.

• Option 2: \(BC = 2\sqrt{3}, AC = 2\)

If \(BC = x\sqrt{3} = 2\sqrt{3}\), then \(x = 2\), so the hypotenuse \(AC = 2x = 4\). Here \(AC = 2\), which is incorrect.

• Option 3: \(AB = 3, AC = 3\sqrt{3}\)

If \(AB = x = 3\), then \(AC = 2x = 6\). Here \(AC = 3\sqrt{3}\), which is incorrect.

• Option 4: \(BC = 10, AC = 4\sqrt{3}\)

If \(BC = x\sqrt{3} = 10\), then \(x = \frac{10}{\sqrt{3}}\), so \(AC = \frac{20}{\sqrt{3}}\). Here \(AC = 4\sqrt{3}\), which is incorrect.

• Option 5: \(AB = 7, AC = 14\)

If \(AB = x = 7\), then \(AC = 2x = 14\). This is possible.

• Option 6: \(AB = 11, BC = 11\sqrt{3}\)

If \(AB = x = 11\), then \(BC = x\sqrt{3} = 11\sqrt{3}\). This is possible.

Select the three correct options

The three possible pairs of measurements are:

1. \(AB = 4, BC = 4\sqrt{3}\)
2. \(AB = 7, AC = 14\)
3. \(AB = 11, BC = 11\sqrt{3}\)

Answer:

• \(AB = 4, BC = 4\sqrt{3}\) (Correct answer)
• \(BC = 2\sqrt{3}, AC = 2\)
• \(AB = 3, AC = 3\sqrt{3}\)
• \(BC = 10, AC = 4\sqrt{3}\)
• \(AB = 7, AC = 14\) (Correct answer)
• \(AB = 11, BC = 11\sqrt{3}\) (Correct answer)