12. If G is the circumcenter of triangle ACE, Find GD

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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