Question

question 1 of 10
which set of values could be the side lengths of a 30 - 60 - 90 triangle?
a. {6,6√3,12}
b. {6,12,12√2}
c. {6,12,12√3}
d. {6,6√2,12}

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a 30-60-90 triangle, the side lengths are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is the shortest (\(x\)), opposite 60° is \(x\sqrt{3}\), and opposite 90° (hypotenuse) is \(2x\).

Step2: Analyze Option A

For set \(\{6, 6\sqrt{3}, 12\}\), let \(x = 6\). Then \(x\sqrt{3}=6\sqrt{3}\) and \(2x = 12\). This matches the \(1 : \sqrt{3} : 2\) ratio.

Step3: Analyze Option B

Set \(\{6, 12, 12\sqrt{2}\}\) has a ratio \(6:12:12\sqrt{2}=1:2:2\sqrt{2}\), which is the ratio for a 45-45-90 triangle (\(1:1:\sqrt{2}\) scaled), not 30-60-90.

Step4: Analyze Option C

Set \(\{6, 12, 12\sqrt{3}\}\) has ratio \(6:12:12\sqrt{3}=1:2:2\sqrt{3}\), which does not match 30-60-90 ratio.

Step5: Analyze Option D

Set \(\{6, 6\sqrt{2}, 12\}\) has ratio \(6:6\sqrt{2}:12 = 1:\sqrt{2}:2\), which is 45-45-90 triangle ratio (scaled), not 30-60-90.

Answer:

A. \(\{6, 6\sqrt{3}, 12\}\)