We are asked to find the length of DF. Let's consider the diagram.
As we can see, DG consists of two segments: DF and FG. Therefore, by the Segment Addition Postulate, its length equals the sum of these segment lengths.
DG= DF+FG
The length of DG is known from the diagram. How can we find the length of DF? From the diagram, we see that HD and HG have the same length. This means point H is equidistant from D and G. Let's now use the Converse of the Perpendicular Bisector Theorem.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
According to the theorem, HF is the perpendicular bisector of DG. Therefore, point F is the midpoint of DG. We conclude that the length of DF is the same as the length of FG. Finally, we can substitute FG with DF in the equation and solve for DF.
