Question

The points (2, -1), (-1, 4) and (-2, 2) are midpoints of the sides ofa triangle. Find its vertices.

Answer 1

Let A(x₁,y₁), B(x_2, y₂ ) and C(x₃, y₃) be the coordinates of the vertices of Δ ABC.

Midpoint of AB, i.e. D 

D (2 , 1) = D `(("x"_1 + "x"_2)/2 , ("y"_1 + "y"_2)/2)`

`2 = ("x"_1 + "x"_2)/2 , ("y"_1 + "y"_2)/2` = -1

X₁ + X₂ = 4 .....(1)                Y₁ + Y₂ = -2  ........(2)

Similarly, 

X₁ + X₃ = -2  .....(3)              y₁ + y₃ = 8  . ....(4) 

X₂ + X₃ = -4   .....(5)             Y₂ + Y₃ = 4  .....(6)  

Adding (1), (3) and (5) 

2(x₁ + x₂ + x₃) = -2 

x₁ + x₂ + x₃  = -1

4 + x₃ = -1  ... (from (1))

x₃  = -5

From (3)

x₁ - 5 = -2

x₁  = 3

From (5)

x₂ = -5 = -4

x₂ = 1

Adding (2),(4) and (6) 

2( Y₁ + Y₂ + Y₃)= 10 

Y₁ +Y₂ +Y₃ = 5 

-2 + Y₃ = 5      [from(2)] 

y₃ = 7 

From (4)

y₁ + 7 = 8

y₁ = 1 

From (6)

y₂ + 7 = 4

y₂ = -3

The coordinates of the vertices of A.ABC are (3, 1), ( 1,-3) and ( -5,7)