Centroids divide a median (like AD) into two segments that are in a 2:1 ratio. That's the centroid's special power. So if AD is a total of 15 units long, then dividing it into two segments in a 2:1 ratio means the long side is 10 and the short side is half that, or 5, which checks out because 10+5 = 15.
Now you have GD = 5, and you also know that angle GDC is 90 degrees, so you know triangle GDC is a right triangle, which means it obeys the Pythagorean Theorem.
