How to solve for the centroid of A(4,2), B(12,6) and C(7,13)
The three points make up a triangle. We need to find the central points of each side.
This helps us to establish the medians from each corner of the triangle to the midpoint of the opposite side.
The centroid will be located where the medians intersect.
The sides of the triangle are AB with midpoint P; BC with midpoint Q; CA with midpoint R.
We only need two medians to establish the centroid, but it would be wise to use the third median as a double-check.
We now work out the equations of the medians.
Join P(8,4) to corner C(7,13): slope=(13-4)/(7-8)=-9. Intercept=4+9*8=76, and y=76-9x (using P to subsitute in equation y=-9x+b, where b is intercept).
Join Q(9.5,9.5) to corner A(4,2): slope=(2-9.5)/(4-9.5)=+7.5/5.5=15/11. Intercept=2-15/11*4=-38/11 and y=1/11*(15x-38).
Join R(5.5,7.5) to corner B(12,6): slope=(6-7.5)/(12-5.5)=-3/13. Intercept=6+3/13*12=114/13 and y=1/13(114-3x).
Equating these three equations: 76-9x=1/11(15x-38)=1/13(114-3x) we can solve for x.
76-9x=836-99x=15x-38, so 114x=874. x=874/114=23/3 and y=7.
The centroid is (23/3,7).
Also, 76-9x=1/13(114-3x), so 988-117 x=114-3x and 874=114x. x=23/3 and y=7, confirming the position of the centroid. Read More: ...