Explanation:
there are some formulas to know about this.
I am sure you discussed this in class.
when 2 lines (originating from the same point) cut through the same circle creating some circle segments (such lines are also called Secants), then there is a relationship in the lengths of the different parts of the lines among themselves.
the outer portion of such a line multiplied with the length of the whole line to the second point it intersects the circle is the same for every secant from the same point through the same circle.
that means in our case :
7(x - 6) = 6(x - 3)
7x - 42 = 6x - 18
x = 24
and so,
EG = (x - 3) + 6 = (24 - 3) + 6 = 21 + 6 = 27