Thank you for the opportunity to help you with your question!

It can be solved using the following theorem which is secant theorem

"If two secant segments are
drawn to a circle from the same external point, the product of
the length of one secant segment and its external part is equal
to the product of the length of the other secant segment and its
external part."

Here the same point is G.

Using the above theorem

GH*DG=FG*EG

We have DG=x+3+5=x+8

GH=5

FG=6

GE=x+6

Thus

(x+8)5 =6(x+6)

Solve for x

5x+40=6x+36

x=4

We have DG=x+8 =4+8=12(answer)

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