Question

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

a) 6 in
b) 9 in
c) 12 in
d) 15 in

Answer 1

CG, the segment from vertex C to the centroid G in right triangle ABC with hypotenuse AB measuring 18 inches, is 6 inches long, because the centroid divides the median from C to the midpoint of the hypotenuse in a 2:1 ratio.

Step-by-step explanation:

To find the length of segment CG when G is the centroid of right triangle ABC with hypotenuse AB measuring 18 inches, we recall the property of the centroid of a triangle. The centroid divides each median in a ratio of 2:1, with the larger segment being closer to the corresponding vertex. Since the median CG goes from vertex C to the midpoint of the hypotenuse AB, and the centroid divides this median into a 2:1 ratio, the length of CG is two-thirds the length of the median CB. As AB is the hypotenuse of the triangle and measures 18 inches, the median CB, being the segment from a right-angle vertex to the midpoint of the hypotenuse, will be half that length, so CB = 18 inches / 2 = 9 inches. Therefore, CG = 2/3 × 9 inches = 6 inches. Hence, the correct answer is option a) 6 inches.