Question

if the co ordinate of mid points of side AB, BC , CA of triangle ABC are(1,1), ( 2,-3) and (3,4) then find co ordinates of centroids

Answer 1

According to given sum,

let the co-ordinates of
vertex A be (x1,y1)
vertex B be (x2,y2)
vertex C be (x3,y3)

given that mid-point of AB is (1,1)

((x1+x2)/2, (y1+y2)/2)=(1,1)

comparing the x coordinates & y co ordinates
(x1+x2)/2=1 (y1+y2)/2=1
x1+x2=2...........(i) y1+y2=2............. (ii)


Also given that mid point of AC is (3,4)

((x1+x3)/2, (y1+y3)/2)=(3,4)

comparing the x coordinates & y coordinates
(x1+x3)/2=3 (y1+y3)/2=4
x1+x3=6.......... (iii) y1+y3=8............ (iv)

subtracting equation i & ii and ii & iv

x1+x2-x1-x3=2-6 y1+y2-y1-y3=2-8
x2-x3=-4 .......... (v) y2-y3=-6............. (vi)

given that mid point of BC is (2,-3)

((x2+x3)/2, (y2+y3)/2)=(2,-3)

comparing the x & y coordinates
(x2+x3)/2=2 (y2+y3)/2=-3
x2+x3=4.........(vii) y2+y3=-6........... (viii)

Adding equation v & vii and vi & viii
x2-x3+x2+x3=-4+4 y2-y3+y2+y3=-6-6
x2=0/2 y2=-12/2
x2=0 y2=-6

so point B is (0,-6) =(x2,y2)
substituting this value in equations i & ii
x1+x2=2 y1+y2=2
x1+0=2 y1-6=2
x1=2 y1=8

so point of A is (2,8)

substituting this values in equation iii & iv


x1+x3=6 y1+y3=8
2+x3=6 8+y3=8
x3=4 y3=0

so point of C is (4,0)

centroid of the triangle is
((x1+x2+x3)/2, (y1+y2+y3)/2)
((2+0+4)/2, (8-6+0)/2)
(6/2,2/2)
(3,1)

:-)Hence the answer is (3,1)

Hope it helps u.

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