Special Right Triangles

I’m struggling to understand the sources of the square roots found on the special right triangles? How do we decide on the square roots and such?

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Hey @Megan, welcome to the Achievable community.

There are three main types of special right triangles, and you just need to memorize them. Given two of the angles of a triangle, you can calculate the third angle, since they will always add up to 180 degrees. And then given any one of the sides, you can calculate the other two sides.

Here’s a quick summary:

  1. 30-60-90 - Sides are x, x*sqrt(3), and 2*x. The image you sent gave us the angles of 30-60-90 and the hypotenuse (long side) of 4. Using the pattern, we know that 4 is actually the side of length 2*x, so that means x=2, and the other sides are 2 and 2*sqrt(3).

  2. 45-45-90 - Sides are x, x, and x*sqrt(2). The hypotenuse of a triangle with these angles will have a length of x*sqrt(2). Our hypotenuse is given as 4, so we can divide by sqrt(2) and figure out that x is actually 4/sqrt(2).

  3. Pythagorean triplets

These triangles have lengths that just happen to make nice right angles. Any multiple of these numbers will also be a right triangle.

  • 3-4-5
  • 5-12-13
  • 6-8-10 (i.e. 3-4-5 multiplied by two)
  • 10-24-26 (i.e. 5-12-13 multiplied by two)
  • 9-12-15 (i.e. 3-4-5 multiplied by three)
  • 15-36-39 (i.e. 5-12-13 multiplied by three)

Hope this clears it up - you can read more in the Right triangles | Triangle problems chapter!