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?
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:

306090  Sides are x, x*sqrt(3), and 2*x. The image you sent gave us the angles of 306090 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).

454590  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).

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.
 345
 51213
 6810 (i.e. 345 multiplied by two)
 102426 (i.e. 51213 multiplied by two)
 91215 (i.e. 345 multiplied by three)
 153639 (i.e. 51213 multiplied by three)
Hope this clears it up  you can read more in the Right triangles  Triangle problems chapter!