Question

Point G is the centroid of the right △ABC with m∠C=90° and m∠B=30°. Find AG if CG=4 ft.

Answer 1

Answer: $\text{Length of AG=}\frac{2\sqrt{63}}{3}$

Explanation:  

Please follow the diagram in attachment.  

As we know median from vertex C to hypotenuse is CM  

$\therefore CM=\frac{1}{2}AB$

We are given length of CG=4  

Median divide by centroid 2:1  

CG:GM=2:1  

Where, CG=4

$\therefore GM=2$ ft

Length of CM=4+2= 6 ft  

$\therefore CM=\frac{1}{2}AB\Rightarrow AB=12$

In $\triangle ABC, \angle C=90^0$

Using trigonometry ratio identities  

$AC=AB\sin 30^0\Rightarrow AC=6$ ft

$BC=AB\cos 30^0\Rightarrow BC=6\sqrt{3}$ ft  

$CN=\frac{1}{2}BC\Rightarrow CN=3\sqrt{3}$ ft

In $\triangle CAN, \angle C=90^0$  

Using pythagoreous theorem  

$AN=\sqrt{6^2+(3\sqrt{3})^2\Rightarrow \sqrt{63}$

Length of AG=2/3 AN

$\text{Length of AG=}\frac{2\sqrt{63}}{3}$ ft