The length of FH is 16 units.
In triangle FGH, GJ is an angle bisector of angle G and perpendicular to FH. We are given that FG = 3x - 8, FJ = x, and GH = 16. We need to find the length of FH.
Since GJ is an angle bisector of angle G, it divides angle G into two congruent angles, ∠FJG and ∠HJG. Therefore, triangles FJG and HJG are right triangles and share a hypotenuse, GJ.
We can use the Pythagorean theorem to find the lengths of FJ and FH. In triangle FJG, we have:
FJ^2 + JG^2 = FG^2
Substituting the given values, we get:
x^2 + JG^2 = (3x - 8)^2
Solving for JG, we get:
JG = √(14x^2 - 48x + 64)
Similarly, in triangle HJG, we have:
JH^2 + JG^2 = GH^2
Substituting the given values, we get:
JH^2 + √(14x^2 - 48x + 64)^2 = 16^2
Solving for JH, we get:
JH = √(256 - 14x^2 + 48x - 64)
Since FH is the hypotenuse of right triangles FJG and HJG, we can find its length using the Pythagorean theorem:
FH^2 = FJ^2 + JH^2
Substituting the expressions we obtained for FJ and JH, we get:
FH^2 = x^2 + √(14x^2 - 48x + 64)^2 + √(256 - 14x^2 + 48x - 64)^2
Simplifying the expression, we get:
FH^2 = 256
Therefore, the length of FH is 16 units.