Question

the mid points of the sides AB,BC,CA of a triangle ABC are D(2,1),E(1,0) and F(-1,3) respectively.Find the coordinates of the vertices of the triangle ABC?

Answer 1

given,
A(a,b) B(c,d) C(e,f)
Midpoint of AB:
(2,1)=(a+c/2),(b+d/2)
a+c=4........(1) and b+d=2.......(2)
Midpoint of BC:
(1,0)=(c+e/2),(d+f/2)
c+e=2........(3) and d+f=0.........(4)
Midpoint of AC:
(-1,3)=(a+e/2),(b+f/2)
a+e=-2.........(5) and b+f=6.........(6)
(1)-(3)=>
a+c-c-e=4-2
a-e=2........(7)
(5)+(7):
a+e+a-e=-2+2
2a=0
a=0
substitute a=0 in (1)
c=4
substitute a=0 in (5)
e=-2
(2)-(4)=>
b+d-d-f=2-0
b-f=2..........(8)
(6)+(8):
b+f+b-f=6+2
2b=8
b=4
substitute b=4 in (2)
4+d=2
d=-2
substitute d=-2 in (4)
-2+f=0
f=2

A(0,4) B(4,-2) C(-2,2)

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