30-60-90 Triangles

An equilateral triangle is a triangle with all sides congruent.  From the Isosceles Triangle Theorem if follows that the angles of an equilateral triangle are all the same.  Since the angles of any triangle add to 180^o, each angle of an equilateral triangle must be 60^o.  When an equilateral triangle is cut in half, you get two right triangles.  The acute angles of these triangles are 30^o and 60^o, so we call them "30-60-90 triangles": 

If a side of the equilateral triangle is 2 units in length, then the side opposite the 30^o angle in one of the 30-60-90 triangles is half that, or 1 unit, and the third side (the side opposite the 60^o angle) can be found by the Pythagorean Theorem:

In general, in a 30-60-90 triangle, the side opposite the 30^o angle is half the hypotenuse.  The other side can then be found by the Pythagorean Theorem, and will always be a multiple of the square root of 3.  More specifically, let the length of the hypotenuse be 2a.  Then the side opposite the 30^o angle will be of length a, and we can find the third side as follows:

Notice that we can simplify  by separating into the product of two square roots:

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