Answer: [tex]\text{Length of AG=}\frac{2\sqrt{63}}{3}[/tex]
Explanation:
Please follow the diagram in attachment.
As we know median from vertex C to hypotenuse is CM
[tex]\therefore CM=\frac{1}{2}AB[/tex]
We are given length of CG=4
Median divide by centroid 2:1
CG:GM=2:1
Where, CG=4
[tex]\therefore GM=2[/tex] ft
Length of CM=4+2= 6 ft
[tex]\therefore CM=\frac{1}{2}AB\Rightarrow AB=12[/tex]
In [tex]\triangle ABC, \angle C=90^0[/tex]
Using trigonometry ratio identities
[tex] AC=AB\sin 30^0\Rightarrow AC=6[/tex] ft
[tex] BC=AB\cos 30^0\Rightarrow BC=6\sqrt{3}[/tex] ft
[tex] CN=\frac{1}{2}BC\Rightarrow CN=3\sqrt{3}[/tex] ft
In [tex] \triangle CAN, \angle C=90^0[/tex]
Using pythagoreous theorem
[tex]AN=\sqrt{6^2+(3\sqrt{3})^2\Rightarrow \sqrt{63}[/tex]
Length of AG=2/3 AN
[tex]\text{Length of AG=}\frac{2\sqrt{63}}{3}[/tex] ft
