Question

Find height of triangle in a 45-45-90 triangle knowing one side

Answer 1

Look at the picture.
It's the isosceles triangle.
Use the Pythagorean theorem:
$x^2=a^2+a^2\\\\x^2=2a^2\\\\x=\sqrt{2a^2}\\\\x=a\sqrt2$
Therefore:
If you are knowing the length of side a, then $x=a\sqrt2$.
If you are knowing the length of side x, then $a=\dfrac{x\sqrt2}{2}$