Question

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 triangle properties

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

• Short leg opposite \( 30^\circ \): \( AB = x \)
• Long leg opposite \( 60^\circ \): \( BC = x\sqrt{3} \)
• Hypotenuse opposite \( 90^\circ \): \( AC = 2x \)

Evaluate each option

• Option 1: \( AB = 4, BC = 4\sqrt{3} \). Here \( x = 4 \), so \( BC = x\sqrt{3} = 4\sqrt{3} \). This is possible.
• Option 2: \( BC = 2\sqrt{3}, AC = 2 \). Here \( x\sqrt{3} = 2\sqrt{3} \implies x = 2 \), which means \( AC = 2x = 4

e 2 \). This is impossible.

• Option 3: \( AB = 3, AC = 3\sqrt{3} \). Here \( x = 3 \), which means \( AC = 2x = 6

e 3\sqrt{3} \). This is impossible.

• Option 4: \( BC = 10, AC = 4\sqrt{3} \). This does not satisfy the ratio \( AC = \frac{2}{\sqrt{3}} BC \). This is impossible.
• Option 5: \( AB = 7, AC = 14 \). Here \( x = 7 \), so \( AC = 2x = 14 \). This is possible.
• Option 6: \( AB = 11, BC = 11\sqrt{3} \). Here \( x = 11 \), so \( BC = x\sqrt{3} = 11\sqrt{3} \). This is possible.

Select the three correct pairs

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:

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