Question

In triangle GHJ, K (3,9) is the midpoint of segment GH, L (12,3) is the midpoint of HJ, and M (15,6) is the midpoint of segment GJ. Find the coordinates of G, H, and J

Answer 1

The coordinates of G, H, and J include the following:

G (6, 12)

H (0, 6)

J (24, 0)

How to determine the coordinates of the other endpoints?

In order to determine the midpoint of a line segment with two (2) end points, we would add each end point together and then divide by two (2):

`Midpoint=[(x_1+x_2)/(2) ,(y_1+y_2)/(2) ]`

Next, we would determine the coordinates of G, H, and J by substituting our given end points as follows;

`(x_K,y_K)=((x_G + x_H)/(2) ,(y_G + y_H)/(2) )\\\\(3,9)=((x_G + x_H)/(2) ,(y_G + y_H)/(2)) \\\\(12,3)=((x_H + x_J)/(2) ,(x_H + x_J)/(2) )\\\\(15,6)=((x_G + x_J)/(2) ,(y_G + y_J)/(2))`

Based on the system of equations, we have the following:

`x_G = 6 - x_H\\\\x_J = 24 - x_H\\\\y_G = 18 - y_H\\\\y_J = 6 - y_H`

By using the substitution method, we have:

`15 = ((6 - x_H + 24 - x_H))/(2) \\\\15 = ((30 - 2x_H))/(2) \\\\30=30 - 2x_H\\\\x_H=0`

`6 = ((18 - y_H + 6 - y_H) )/(2)\\\\12=24-2y_H\\\\2y_H=12\\\\y_H=6`

Therefore, the coordinates of H are (0, 6).

Next, we would determine the coordinates of points G and J as follows;

`x_G = 6 - x_H\\\\x_G = 6 - 0 = 6\\\\\\y_G = 18 - y_H\\\\y_G = 18 - 6 = 12\\\\\\\\x_J = 24 - x_H\\\\x_J = 24 - 0 = 24\\\\\\y_J = 6 - y_H\\\\y_J = 6 - 6 = 0`