Determine algebraically the coordinates of the a)centroid b)circumcener

Question

Triangle DEF has the following properties:
Vertice D(6,3) vertice E(1,-2) vertice F(4,-1)
Median : from D: y=9/7x-33/7
median : from E : y=3/4x-11/4
Median : from F: y=-3x+11
Perpendicular bisector of DE:y=-x+4
perpendicular bisector of EF:y=-3x+6
perpendicular bisector of DF:y=-1/2x+7/2
Determine algebraically the coordinates of the a)centroid b)circumcener

Answer 1

the centroid is where the medians intersect

the 3 equations of the 3 medians have one point in common-the centroid

pick any 2 medians and find what point they have in common-this will be the centroid

let's use y=(3/4)x-(11/4) and y=-3x+11

equate the right hand sides

(3/4)x-(11/4)=-3x+11

(3/4)x+3x=11+(11/4)

(3 3/4)x=(44/4)+(11/4)

(15/4)x=(55/4)

multiply both sides by (4/15)

x=(4/15)(55/4)

x=55/15

x=11/3

now find y by substituting into either equation

y=-3(11/3)+11

y=-11+11

y=0

(11/3, 0) is the centroid

this point also is a point on the third line y=(9/7)x-(33/7)

0=(9/7)(11/3)-(33/7)

0=(99/21)-(33/7)

0=(99/21)-(99/21)

0=0

the circumcenter is where the perpendicular bisectors of the sides intersect

apply the same reasoning as above

pick 2 equations and find their solution-this will be the circumcenter

y=-3x+6 and y=-x+4

-3x+6=-x+4

6-4=3x-x

2=2x

x=1

substitute

y=-3(1)+6

y=-3+6

y=3

(1,3) is the circumcenter