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In a 30degree 60degree 90degree triangle,the ratio of the length of the hypotenuse to the length of the shorter side is

Note that a 30-60-90 triangle is half of a triangle with equal angles and sides.

This makes the hypotenuse one of the sides of the triangle with the shorter leg as one of the sides. Since that leg is half the hypotenuse, the length is 1/2 as much. That makes the ratio be 2:1.

To say a little more, take the shorter side to be length 1.

This makes the hypotenuse be of length 2.

Since it is a right triangle, we know that 1² + s² = 2².

Since 2² = 4 and 1² = 1, that is 1 + s² = 4, so s = √3.

That is why I first remembered the sides of a 30-60-90 triangle to be as simple as 1-2-3.

What to remember is that the lengths are 1-2-√3 with 2 being on the hypotenuse.

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