Dg¯¯¯¯¯¯ , eg¯¯¯¯¯ , and fg¯¯¯¯¯ are perpendicular bisectors of the sides of △abc . dg=5 cm and bd=12 cm. what is cg ?

The answer is that CG is equal to 13cm.
As given that DG, EG, and FG are perpendicular bisectors of the sides of triangle ABC and it shows that G is the point of intersection and also it is the circumcenter of the triangle, So;
AG = BG = CG
By Pythagorean theorem
BG² = DG² + BD²BG²
And DG = 5cm and BD = 12cm
So,
BG = √(DG² + BD²) = √(5² + 12²) = √(25 + 144) = √169 = 13
BG = 13 = AG = CG
Thus, CG = 13cm