
If G is the circumcenter of triangle ACE
Distances between the circumcenter G and the vertices A, E, and C of a triangle are equal.
It means,
AG=CG=EG
Substitute the AG=7x-41, CG=5x-19 into the above expression:
7x-41=5x-19
Collect like terms:
7x-5x=41-19
Simplify The above expression
2x=22
Both sides are divided into 2:
x=\frac{\cancel{22}}{\cancel{2}}=11
Find value of AG=7x-41 for x=11:
AG=7\times11-41
After simplifying we get,
AG=77-41=36
We know that from above AG=CG=EG
AG=CG=EG=36
Here, GD, BG, and GF are perpendicular bisectors of the circumcenter G:
triangle GDE is a right Triangle whose side ED=28:
Let be GD denoted by x:
by using the Pythagorean Theorem:
(GD)^2+(ED)^2=EG^2
Substitute the EG=36, ED=28 into the above
(GD)^2+(28)^2=(36)^2
After simplifying we get
(GD)^2=1296-784=512
The value of GD is:
GD=\sqrt{512}=16\sqrt{2}=22.62
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