Triangle ABC is a right triangle with a short side of \(2 \sqrt{3}\) cm and a hypotenuse of \(4 \sqrt{3}\) cm. Because the short side is half the hypotenuse, this is a 30 °-60 °-90 ° triangle with the 30 ° angle at vertex A. The ratio of the sides of this triangle are \(1:\sqrt{3}:2\) , so side AB is \(\sqrt{3} * 2 \sqrt{3} =6\).
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A perpendicular from B to line AC creates two more 30 °-60 °-90 ° triangles. In the triangle on the left, the perpendicular is the short side and the hypotenuse is 6. The length of the perpendicular is half the length of the hypotenuse, or 3.