In angle def , df=22 and m angle f=45. find the length of a leg. leave your answer in simplest radical form

The figure is the right triangle DEF.

In it, E is the 90° angle, D is the corner over E (this is, DE is the vertical segment or height from the base) and F is the corner to the right of E (f, is the angle and EF is the horizontal segment or base of the triangle).

Then EF is the adjacent leg to the angle f, DE is the opposite leg to the angle f, and DF is the hypotenuse of the triangle.

Given that the angle f is 45°, you know that the triangle is isosceles (45-45-90) and then the two legs have the same length and you can use Pytagorean theorem:

DF^2 = x^2 + x^2 = 2x^2 => x^2 = (DF)^2 / 2

Taking square roots from both sides:

=> x =  DF / √2 = 22 / √2 = [22√2] / [(√2)(√2)] = [22√2] / 2 = 11√2

Remember, both legs have the same length.

11√2

This equation is not factor-able, so we can use the quadratic formula.
x=(-11+square root(11^2-4(-12*-3)))/(2*-12)
x=(-11+square root(-23))/(-24)
Since we have the square root of a negative in this equation, there are no real roots for the quadratic.