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I tutor students in algebra, geometry, and test prep math, and this is my ode to special right triangle side ratios. I wanted to make something catchy and upbeat, since many students have a hard time remembering the side ratios that belong to each triangle.

Lyrics:
x, 2x, and x root 3 for the 30° - 60° - and 90°
x, 2x, and x root 3 for the 30° - 60° - and 90°

2x is the longest side...I just had to make it rhyme
the longest side is the hypotenuse,
now we can introduce…

x, x, and x root 2...congruent sides? we’ve got 2!
x, x, and x root 2...congruent sides? we’ve got 2!

45° - 45° - 90°...an isoceles righty...
x root 2 is the hypotenuse,
no excuse to be obtuse…

Examples:
If you have a 45° - 45° - 90° triangle, and the legs (across from the 45° angle) = 6, then the hypotenuse = 6√2.

If you have a 30° - 60° - 90° triangle, and the shortest side (across from the 30° angle) = 2, then the side across from the 60° angle = 2√3, and the hypotenuse = 4.

If the hypotenuse of your 45° - 45° - 90° triangle is 8, then divide by √2 to find the lengths of the legs. They would = 8/√2, rationalize to get (8√2)/2 and reduce to get 4√2.

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