The length of GH is 3.
To find the length of GH, we can use the Angle Bisector Theorem. This theorem states that the ratio of the side lengths opposite the angles formed by an angle bisector is equal to the ratio of the side lengths adjacent to those angles.
In other words, we have the following proportion:
(EG / GH) = (EF / FH)
We know that EG = 12x-3 and EF = 8x+5. We also know that FH = GH + 12x-3 (since GH + EG = FH).
Substituting these values into the proportion, we get:
(12x-3 / GH) = (8x+5 / (GH + 12x-3))
Multiplying both sides by GH + 12x-3, we get:
12x-3 = (8x+5) * GH
Expanding the right-hand side, we get:
12x-3 = 8xGH + 5GH
Combining like terms, we get:
4xGH = 12
Dividing both sides by 4x, we get:
GH = 3
Therefore, the length of GH is 3.