If D(3,-2) , E(-3,1) and F(4,-3) are the mid-points of the sides BC, CA and AB respectively of `Delta ABC` , find the co-ordinates of point A , B and C .

Answer

To find the coordinates of points A, B, and C given the midpoints D(3, -2), E(-3, 1), and F(4, -3) of the sides BC, CA, and AB respectively of triangle ABC, we can follow these steps:

### Step 1: Set up the equations using the midpoint formula
Let the coordinates of points A, B, and C be:
- A(x₁, y₁)
- B(x₂, y₂)
- C(x₃, y₃)

Using the midpoint formula, we can set up the following equations:

1. For midpoint F(4, -3) of AB:
\[ \frac{x₁ + x₂}{2} = 4 \quad \text{(1)} \]
\[ \frac{y₁ + y₂}{2} = -3 \quad \text{(2)} \]

2. For midpoint E(-3, 1) of AC:
\[ \frac{x₁ + x₃}{2} = -3 \quad \text{(3)} \]
\[ \frac{y₁ + y₃}{2} = 1 \quad \text{(4)} \]

3. For midpoint D(3, -2) of BC:
\[ \frac{x₂ + x₃}{2} = 3 \quad \text{(5)} \]
\[ \frac{y₂ + y₃}{2} = -2 \quad \text{(6)} \]

### Step 2: Solve the equations
From equations (1) and (2):
- From (1):
\[ x₁ + x₂ = 8 \quad \text{(7)} \]
- From (2):
\[ y₁ + y₂ = -6 \quad \text{(8)} \]

From equations (3) and (4):
- From (3):
\[ x₁ + x₃ = -6 \quad \text{(9)} \]
- From (4):
\[ y₁ + y₃ = 2 \quad \text{(10)} \]

From equations (5) and (6):
- From (5):
\[ x₂ + x₃ = 6 \quad \text{(11)} \]
- From (6):
\[ y₂ + y₃ = -4 \quad \text{(12)} \]

### Step 3: Find x-coordinates
Now, let's solve for the x-coordinates:
1. Add equations (7), (9), and (11):
\[ (x₁ + x₂) + (x₁ + x₃) + (x₂ + x₃) = 8 - 6 + 6 \]
This simplifies to:
\[ 2x₁ + 2x₂ + 2x₃ = 8 \implies x₁ + x₂ + x₃ = 4 \quad \text{(13)} \]

2. Substitute \(x₂ + x₃ = 6\) from (11) into (13):
\[ x₁ + 6 = 4 \implies x₁ = -2 \]

3. Substitute \(x₁ = -2\) into (7):
\[ -2 + x₂ = 8 \implies x₂ = 10 \]

4. Substitute \(x₁ = -2\) into (9):
\[ -2 + x₃ = -6 \implies x₃ = -4 \]

### Step 4: Find y-coordinates
Now, let's solve for the y-coordinates:
1. Add equations (8), (10), and (12):
\[ (y₁ + y₂) + (y₁ + y₃) + (y₂ + y₃) = -6 + 2 - 4 \]
This simplifies to:
\[ 2y₁ + 2y₂ + 2y₃ = -8 \implies y₁ + y₂ + y₃ = -4 \quad \text{(14)} \]

2. Substitute \(y₂ + y₃ = -4\) from (12) into (14):
\[ y₁ - 4 = -4 \implies y₁ = 0 \]

3. Substitute \(y₁ = 0\) into (8):
\[ 0 + y₂ = -6 \implies y₂ = -6 \]

4. Substitute \(y₁ = 0\) into (10):
\[ 0 + y₃ = 2 \implies y₃ = 2 \]

### Final Coordinates
Thus, the coordinates of points A, B, and C are:
- A(-2, 0)
- B(10, -6)
- C(-4, 2)