Tenth Grade · Math · week 24

Special right triangles: 45-45-90 and 30-60-90

Apply the fixed side ratios of the 45-45-90 and 30-60-90 triangles to find missing sides directly.

Two right triangles show up so often that their side ratios are worth memorizing outright. A 45-45-90 triangle (an isosceles right triangle, half of a square cut along its diagonal) always has legs of equal length x and a hypotenuse of x * sqrt(2) -- the Pythagorean Theorem applied to two equal legs simplifies to exactly that pattern every time. Once you know one side, the other two follow instantly, no calculator required.

A 30-60-90 triangle (half of an equilateral triangle, split by an altitude from one vertex to the midpoint of the opposite side) has a fixed ratio too: the side opposite the 30-degree angle (the shortest side) is x, the side opposite the 60-degree angle (the longer leg) is x * sqrt(3), and the side opposite the 90-degree angle (the hypotenuse) is 2x. The hypotenuse is always exactly DOUBLE the shortest side -- a quick way to check the sides are identified correctly before applying the ratio.

📋 For the teacher
Teach it (10–15 min):
  • Fold a square along its diagonal to physically produce a 45-45-90 triangle, and cut an equilateral triangle in half with an altitude to produce a 30-60-90 -- seeing WHERE these ratios come from beats memorizing them cold.
  • Keep both ratio patterns on one reference card (45-45-90: x, x, x*sqrt(2); 30-60-90: x, x*sqrt(3), 2x) and drill converting between the three sides in both directions.
  • Stress the "hypotenuse is double the shortest side" check for 30-60-90 triangles as a fast way to catch a mislabeled side before finishing a problem.
  • Show both patterns are really just the Pythagorean Theorem in disguise by verifying one example numerically: a 45-45-90 with legs 5 gives hypotenuse sqrt(50) = 5*sqrt(2).
Talk about it:
  • Where does the sqrt(2) in the 45-45-90 ratio actually come from?
  • Why is the hypotenuse of a 30-60-90 triangle always exactly double the shortest side?
  • If you only know the LONGER LEG of a 30-60-90 triangle, how would you find the other two sides?

Mastery looks like: They correctly apply the 45-45-90 and 30-60-90 side ratios in either direction to find any missing side, without needing the Pythagorean Theorem from scratch.

Common stumbles: Multiplying by sqrt(2) or sqrt(3) when they should divide (going the wrong direction from a given hypotenuse or longer leg); mixing up which special triangle a problem describes.

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