For your second question, not exactly... the side length with length xsqrt(3) *always* must be adjacent to the 30 degree angle! That picture isn't exactly drawn to scale, but with any 30-60-90 triangle, one of the sides (the one with length xsqrt(3)) is always longer than the other. This diagram might help:

Basically, by dropping an altitude from an equilateral triangle, you can see that the shorter side (the one between 60 degrees and the right angle) always has length x. And then you can use the Pythagorean theorem to solve for the blue ?.

Does that make more sense?

]]>So in this case, we have a 30-60-90 triangle, and the hypotenuse is given to be 3. So then we know that the other sides have length (3/2) and (3/2)*sqrt(3).

You definitely don't have to memorize this-- if you understand how the proof for how the lengths are derived (hint: drop an altitude from an equilateral triangle) and use the fact a few times, it becomes pretty intuitive.

For your first question, the height of the new "mail" shape isn't 3 because the segment with line 3 isn't perpendicular to the other diagonal. Remember that the "height" of something always has to form a 90 degree angle with the base! This is basically the same ideas as why, if you have some triangle ABC, drawing any line from A to hit BC isn't necessarily the height of ABC. The height is only the line from A that is perpendicular to BC.

I hope this helped!

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