Answer:- CG= 13 cm
Explanation:-
Given : Δ ABC where DG, EG, and FG are perpendicular bisectors of the sides intersecting at point G .
⇒ G is the circumcenter of the triangle.
⇒BG, CG and AG are the angle bisectors of the triangle.
Now, from the given figure
In ΔBDG and ΔBEG
∠D = ∠ E [right angle]
BG=BG [reflexive property]
∠DBG=∠EBG [Definition of angle bisector]
⇒ΔBDG ≅ ΔBEG [AAS congruence theorem]
Thus BD=BE= 12 cm [CPCT]
DG=EG= 5 cm [CPCT] .....(1)
Now In ΔBGE and ΔCGE
GE=GE [reflexive property]
∠BEG=∠CEG [right angle]
BE=EC [Definition of perpendicular bisector]
⇒ΔBGE ≅ ΔCGE [SAS postulate]
⇒BE=CE=12 cm ......(from (1))
and GE= 5 cm
As ΔCGE is right angle then by Pythagoras theorem,
[tex]CG^2=GE^2+EC^2\\\Rightarrow\ CG=5^2+12^2\\\Rightarrow\ CG^2=25+144=169\\\Rightarrow\ CG=\sqrt{169}=13\ cm[/tex]
