In the xy-coordinate plane, point G(4, 3) is the centroid of triangle

Rule: in any triangle, the median is drawn from a vertex of the triangle such that it bisects the opposite side at the Mid-Point of that opposite side

Rule: for any median drawn from a given vertex, the centroid divides the Median Line in a ratio = 2 : 1

This ratio is the following:

Length from Vertex - to - Centroid : Length from Centroid - to - Midpoint = 2 : 1

For the points given for this triangle, Point A is the Vertex opposite Side BC

Let the Midpoint of Side BC = D

Let the Median drawn from Vertex A to Opposite Side BC (which passes through Centroid Point G) = Median Line AD

Then the Ratio of Lengths along Median Line AD is the following—-

Vertex A to Centroid G : Centroid G to Midpoint D = 2 : 1

1st) The Horizontal Distance from Vertex A to Centroid G is given by the Difference in the respective Points’ X Coordinates

A (1 , 3)
G (4 , 3)

Distance from Vertex A to Centroid G = 4 - 1 = 3

Setting up the Ratios and Proportion equation——-
(Vertex A to Centroid G) / (Centroid G to Midpoint D) = 2/1 = 3/?

Actual Length/Distance from Centroid G to Midpoint D on Side BC = 3/2 = 1.5

1.5 units away from G on the X Axis gives us the X Coordinate of Midpoint D = 5.5

Also, the Median AD is a straight horizontal line, so the Y Coordinate of Midpoint D = same Y Coordinate as Vertex Point A and Point G = 3

Thus, the Midpoint D on Side BC created by Median AD is at the following coordinates:

Midpoint D (5.5 , 3)

Because Vertex Point B has an X coordinate of 4, it must lie somewhere on the vertical line of X = 4

Because Vertex Point C has a Y coordinate of 1, it must lie somewhere on the horizontal line of Y = 1

Based on the relative position of Vertex A, Centroid G, and Midpoint D of side BC——-

Vertex Point B must have a Y Coordinate greater than Centroid G’s Y coordinate of 3—and—-Vertex Point C must have a X Coordinate greater than Centroid G’s X Coordinate of 4

Further, since Median AD Bisects Side BC at Midpoint D:
BD = DC

In order for the above to be true:
The Horizontal Distance from Vertex C (c , 1) to Midpoint D (5.5 , 3) = given by the difference in the X Coordinates = (c - 5.5)

=MUST EQUAL=

The Horizontal Difference from Midpoint D (5.5 , 3) to Centroid G (4, 3) = given by the difference in the X coordinates = (5.5 - 3)

(c - 5.5) = (5.5 - 3)

c = 7

Therefore, Vertex C is at Point (7 , 1)

2nd). Similarly, in order for BD = DC

The Vertical Distance from Vertex Point C (7, 1) to Midpoint D (5.5 , 3) = given by the difference in Y Coordinates = 3 - 1 = 2

=MUST EQUAL=

The Vertical Distance from Midpoint D (5.5, 3) to Vertex B (4 , b) = given by the Difference in the Y coordinates = b - 3

b - 3 = 2
b = 5

Therefore, the Coordinates for Vertex Point B are (4 , 5)

3rd) Lastly, to find the length of side BC, find the Distance from Vertex Point B (4 , 5) to Vertex Point C (7 , 1)

Distance from C to B = sqrt( (5-1)^2 + (4 - 7)^2)

= sqrt(16 + 9) = sqrt(25) = 5

Answer -D-:
Length of Side BC = 5

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