Question 20859: Given the coordinates of the midpoints of the sides of a triangle find the coordinates of the vertices of the triangle.
Midpoint Coordinates:
L(-2,1)
M(2,3)
N(3,-2)
I know that you can plot the coordinates and then draw in the triangle and find the coordinates of the vertices that way, but I was wondering if there was an easier or mathematical way to go about solving the problem. Thank you for looking this over.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Midpoint Coordinates:
L(-2,1)
M(2,3)
N(3,-2)
LET ABC BE THE TRIANGLE WITH COORDINATES OF VERTICES AS A(X1,Y1),B(X2,Y2),C(X3,Y3).LET L,M,N BE THE MIDPOINTS OF AB,BC,CA RESPECTIVELY.AS PER COORDINATE GEOMETRY MIDPOINT'S COORDINATES ARE GIVEN BY THE ARITHMATIC MEAN OF COORINATES OF THE END POINTS..SO..
L={(X1+X2)/2,(Y1+Y2)/2}
HENCE (X1+X2)/2=-2...OR X1+X2=-2*2=-4
X1+X2=-4......(1)
SIMILARLY WE GET FROM M AND N
X2+X3=4.........(2)
X3+X1=6...........(3)
ADDING ALL 3 EQNS.1,2,3,WE GET
X1+X2+X2+X3+X3+X1=2(X1+X2+X3)=-4+4+6=6
X1+X2+X3=3.......(4)
NOW SUBTRACTING EQNS..1,2,AND 3 FROM THIS ,WE GET
X3=7
X2=-1
X1=-3
SIMILARLY YOU CAN FIND Y1,Y2,Y3 WHICH WILL COMPLETE YOUR REQUIREMENT..
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