Question

Let D(3,-2), E(-3,1) and F(4, -3) be the midpoints of the sides
BC, CA and AB respectively of AABC. Then, find the coordinates of
the vertices A, B and C.
​

Answer 1

Given,

D(3,-2), E(-3,1) and F(4, -3) be the midpoints of the sides  BC, CA and AB respectively of ΔABC.

To Find,

coordinates of  the vertices A, B and C.

Solution,

1.

(x₂ + x₃/2) = 3

==> x₂ + x₃ = 6

2.

y₂ + y₃/2 = -2

==> y₂ + y₃ = -4  

3.

(x₃ + x₁)/2 = -3

==> x₃ + x₁ = -6

4.

(y₃ + y₁)/2 = 1

==> y₃ + y₁ = 2

5.

(x₁ + x₂)/2 = 4

==> x₁ + x₂ = 8

6.

(y₁ + y₂)/2 = -3

==> y₁ + y₂ = -6

From (1) and (3).

==> x₂ + x₃ - x₃ - x₁ = 12

==> x₂ - x₁ = 12    ------ (7)

Solve (5) and (7).

2x₂ = 20

==> x₂ = 10

x₁ + x₂ - x₂ + x₁ = 8 - 12

==> 2x₁ = -4

==> x₁ = -2

Place x₁ = -2 in (3).

x₃ + x₁ = -6

==> x₃ - 2 = -6

==> x₃ = -4

Subtract (4) and (2).

y₂ - y₁ = -6    ---- (7)

Solve (7) with (8)

y₂ - y₁ + y₂ + y₁ = -6 - 6

2y₂ = -12

y₂ = -6

Place y₂ in (6).

==> y₁ + y₂ = -6

==> y₁ - 6 = -6

==> y₁ = 0

Place y₁ in (4).

y₃ + 0 = 2

==> y₃ = 2

Result:

Coordinates are A(-2,0), B(10,-6), C(-4,2)