Question

Quickly help : I don’t know WTH I’m doing and my teacher won’t explain

Answer 1

These are what I call the Two Tired Triangles of Trig, 30/60/90 and 45/45/90. If you learn these cold now you'll save yourself some hassle down the road.  These are generally the only right triangles teachers use when they want 'exact' answers.

Here are the facts to remember:

The ratio of the sides of the 45/45/90 triangle are 1:1:√2

The ratio of the sides of the 30/60/90 triangle are 1:√3:2, with the smallest opposite the 30 degree angle.

OK, let's fill in the angles first.  Where we have a right triangle with 30/60/90 we fill in the missing one. For the half squares, the acute angles are 45 degrees.

For the lengths we apply the ratios.  Hypotenuse of 45 degree right triangle is 4, so sides are 4/√2 = 2√2.  When that's opposite 30 degrees, the other two sides are 2√2×√3 = 2√6 and 2×2√2=4√2.  

2√6 is one side of a square, so all sides.  The diagonal is 2√6×√2=2√12=4√3.  

Half the square side is the smallest leg of a 30/60/90, so √6, with other leg √6×√3=√18=3√2 and hypotenuse 2√6.