The coordinates are A(-5,3), B(7,1) and C(-3,-1).
Step-by-step explanation:
Given,
The midpoints of the sides AB, BC and AC are (1,4) , (2,0) and (-4,1) respectively.
To find the coordinates of A, B, C.
Let,
The coordinates of A,B and C are (a,x), (b,y) and (c,z) respectively.
Formula
The mid point of two given points ([tex]x_{1} ,y_{1}[/tex]) and ([tex]x_{2} ,y_{2}[/tex]) is ([tex]\frac{x_{1} +x_{2} }{2}, \frac{y_{1} +y_{2} }{2}[/tex])
Now,
[tex]\frac{a+b}{2}[/tex] = 1 and [tex]\frac{x+y}{2}[/tex] = 4 ⇒ a+b = 2 ---(1) and x+y = 4 --(2)
[tex]\frac{b+c}{2}[/tex] = 2 and [tex]\frac{y+z}{2}[/tex] = 0 ⇒ b+c = 4 ---(3) and y+z = 0---(4)
[tex]\frac{c+a}{2}[/tex] = -4 and [tex]\frac{z+x}{2}[/tex] = 1 ⇒ c+a = -8 ---(5) and z+x = 2 ----(6)
Adding (1), (3) and (5) we get,
2(a+b+c) = -2
or, a+b+c = -1
So, a = -5, b = 7 and c = -3
Similarly Adding (2), (4) and (6) we get,
2(x+y+z) = 6
or, x+y+z = 3
So, x = 3, y = 1 and z = -1
Hence,
The coordinates are A(-5,3), B(7,1) and C(-3,-1).
