Point g is the centroid of the right △abc with m∠c=90° and m∠b=30°. find ag if cg==4 ft.

2√7 ft ≈ 5.2915 ft

A diagram can be helpful.

Let M and N be the midpoints of AB and BC, respectively. Then ∆CMB is an isosceles triangle with 30° base angles. ∆CMA is an equilateral triangle.

Point G is 2/3 of the distance from any vertex to the midpoint of the opposite side for any median. Considering median CM, 2/3 of its length is 4, so its full length must be 4×(3/2) = 6. Then MB = MA = AC = 6. The proportions in 30°-60°-90° triangle ABC are such that CB = 6√3, so CN = 3√3.

By the Pythagorean theorem, ...

... AN² = AC² +CN²

... AN² = 6² +(3√3)² = 63

... AN = √63 = 3√7

So, AG = (2/3)(AN) = 2√7 . . . . . feet

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hope this .. : d