The mid-points of sides AB, BC, CA of triangle ABC are D(2,1), E(1,0), F(-1,3), respectively. Find the coordinates of the vertices of triangle ABC. plz explain step by step(plz answer fast)

The triangle is ABC. 

The mid-points are D (2,1), E (1,0) and F(-1,3) for the sides AB,BC and CA respectively. 

Assuming the coordinates of the points A, B and C as (x₁, y₁), (x₂, y₂) and (x₃, y₃): 

The mid point's coordinates are given by the arithmetic mean of the coordinates of end-points. 
For D it will be: D = [(x₁+ x₂) / 2] = 2 and similarly for E and F. 

So, (x₁+ x₂) = 2 x 2 = 4  ---(1) 

Similarly, (x₂+ x₃) = 2 x 1 = 2   ---(2)
And (x₃+ x₁) = 2 x (-1) = -2    ---(3) 

 

Adding the equations 1, 2 and  3 we get: 2 (x₁+ x_{2 +} x₃) = 4 <==> (x₁+ x_{2 +} x₃) = 2 ---(4)

Subtract equations 1, 2, 3 from 4 to get values of x₁, x₂ and x₃. 
And the values of  x₁, x₂ and x₃ are 0, 4 and -2. 

 

Similarly, we will perform the operation for y-coordinates: 

D: [(y₁+y₂)/2] = 1

Hence, y₁+y₂ = 2 x 1 = 2  ---(5)

Similarly, y₂ + y₃ = 2 x 0 = 0   ---(6)

And y₃ + y₁ = 2 x 3 = 6    ---(7)

Adding equation 5, 6 and 7 we get: 2(y₁ + y₂ + y₃) = 8 <==> (y₁ + y₂ + y₃) = 4   ---(8)

 

Subtract the equations 5, 6 and 7 from 8 to get values of y₁, y₂ and y₃: 
So, the values of y₁, y₂ and y₃ are 4, -2 and 2 respectively. 

 

Hence, (x₁,y₁) = (0, 4)
(x₂, y₂) = (4, -2)

and (x_3, y₃) = (-2, 2)