Question

drag each length to the correct location on the triangle. each length can be used more than once, but not all lengths will be used.
what are the missing side lengths for triangle abc?
8 8\sqrt{2} 4\sqrt{2} 4 12 8\sqrt{3} 4\sqrt{3}
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Explanation:

Step1: Use the properties of a \(30 - 60-90\) triangle

In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). Let the side opposite \(30^{\circ}\) be \(x\), the side opposite \(60^{\circ}\) be \(x\sqrt{3}\), and the hypotenuse be \(2x\). Given that the side opposite \(60^{\circ}\) is \(4\sqrt{3}\).

Step2: Find the side opposite \(30^{\circ}\)

Since the side opposite \(60^{\circ}=x\sqrt{3}\) and \(x\sqrt{3}=4\sqrt{3}\), then \(x = 4\).

Step3: Find the hypotenuse

The hypotenuse \(=2x\). Substituting \(x = 4\), we get \(2\times4=8\).

Answer:

The hypotenuse is \(8\) and the side opposite \(30^{\circ}\) is \(4\).