Answer:
[tex]GD=16\sqrt{2}\ units[/tex]
Step-by-step explanation:
we know that
The three perpendicular bisectors of the sides of a triangle meet in a single point, called the circumcenter (point G)
The circumcenter is equidistant from the vertices of the triangle
so
AG=CG=EG
step 1
Find the value of x
AG=CG
substitute the given values
[tex]7x-41=5x-19[/tex]
solve for x
[tex]7x--5x=41-19[/tex]
[tex]2x=22[/tex]
[tex]x=11[/tex]
step 2
Find the length side DC
we know that
DC=DE
Because GD is a perpendicular bisector
so
[tex]DC=28\ units[/tex]
step 3
Find the length side GD
In the right triangle GCD
Applying the Pythagorean Theorem
[tex]CG^2=GD^2+DC^2[/tex]
we have
[tex]DC=28\ units[/tex]
[tex]CG=5x-19=5(11)-19=36\ units[/tex]
substitute the values
[tex]36^2=GD^2+28^2[/tex]
[tex]GD^2=36^2-28^2[/tex]
[tex]GD^2=512[/tex]
[tex]GD=\sqrt{512}\ units[/tex]
simplify
[tex]GD=16\sqrt{2}\ units[/tex]
