What can be ratios between the lengths of the two legs of a 30-60-90 triangle?

Question

mathstudent55 · Answer

Instead of just memorizing it, just figure it out this way:

Start with an equilateral triangle. It has three 60-degree angles.
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mathstudent55 · Answer

Drop a perpendicular from a vertex to the opposite side. You have now two 30-60-90 triangles. Let the length of the sides of the original equilateral triangle be 2.
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mathstudent55 · Answer

Now use the Pythagorean Theorem to find the length of the altitude, which is a side of the 30-60-90 triangle.

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mathstudent55 · Answer

$a^2 + b^2 = c^2$

$b^2 = c^2 - a^2 = 2^2 - 1^2 = 4 - 1 = 3 $

$b = \sqrt{3} $

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