Question

Point G is the centroid of the right △ABC with hypotenuse AB=18 in. Find CG. 75 points

Answer 1

CG = 6in.

Step-by-step explanation:

The centroid of a triangle, G, is 2/3 way down one of its median. To find CG, we need to find the length of median C.

To find length of median at C, we need Thale's Theorem. It states that a right triangle ABC can be inscribed in a semi-circle with its hypotenuse as a diameter of the circle.

So median C is a point on the semi-circle connecting to mid-point of the hypotenuse AB which is the diameter. Median C connects to the center which makes it a radius. Its length is half of diameter = 18/2 = 9in.

CG is 2/3 way down the median C = 9*2/3 = 6in.

Answer 2

CG = 6 in.

Step-by-step explanation:

Let point M be the midpoint of hypotenuse AB. The vertex C of ∆ABC will lie on the semicircle of diameter AB centered at M. Thus the distance from C to M is ...

... CM = (18 in)/2 = 9 in.

CM is a median of ∆ABC. The centroid of any triangle is at the intersection of medians, which is 1/3 the distance along the median from the side to the vertex. That is, the point G will be 2/3 the distance from C to M.

... CG = (2/3)×(9 in)

... CG = 6 in

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Comment on the problem

Thank you for an interesting question.