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

Question

asked Oct 17, 2019 6.0k views

1 Answer

Daniel Gerson · Answer 1 · 2019-10-22T07:30:52+0000

Answer:
$\text{Length of AG=}(2√(63))/(3)$

Explanation:

Please follow the diagram in attachment.

As we know median from vertex C to hypotenuse is CM

$\therefore CM=(1)/(2)AB$

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

$\therefore CM=(1)/(2)AB\Rightarrow AB=12$

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

Using trigonometry ratio identities

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

$BC=AB\cos 30^0\Rightarrow BC=6√(3)$ ft

$CN=(1)/(2)BC\Rightarrow CN=3√(3)$ ft

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

Using pythagoreous theorem

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

Length of AG=2/3 AN

$\text{Length of AG=}(2√(63))/(3)$ ft