If the coordinates of the mid points of the sides of a triangle are (1,2) , (0,-1) and (2,-1) . Find the coordinates of its vertices :

Options

A. a. (1,-4) , (3,2) , (-1,2)
B. b. (1,2) (2,3) ,(3,-4)
C. c. (3,4), (5,2) , (1,2)
D. d. none of these

Answer

To find the coordinates of the vertices of a triangle given the midpoints of its sides, we can follow these steps:

### Step 1: Define the Midpoints
Let the midpoints of the sides of the triangle be:
- Midpoint M1 = (1, 2) which is the midpoint of side AB
- Midpoint M2 = (0, -1) which is the midpoint of side BC
- Midpoint M3 = (2, -1) which is the midpoint of side AC

### Step 2: Set Up the Equations
Let the vertices of the triangle be:
- Vertex A = (x1, y1)
- Vertex B = (x2, y2)
- Vertex C = (x3, y3)

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

1. For midpoint M1 (AB):
\[ \frac{x1 + x2}{2} = 1 \quad \text{(1)} \]
\[ \frac{y1 + y2}{2} = 2 \quad \text{(2)} \]

2. For midpoint M2 (BC):
\[ \frac{x2 + x3}{2} = 0 \quad \text{(3)} \]
\[ \frac{y2 + y3}{2} = -1 \quad \text{(4)} \]

3. For midpoint M3 (AC):
\[ \frac{x1 + x3}{2} = 2 \quad \text{(5)} \]
\[ \frac{y1 + y3}{2} = -1 \quad \text{(6)} \]

### Step 3: Solve the Equations
From equations (1) and (2):
- From (1): \( x1 + x2 = 2 \) (Equation A)
- From (2): \( y1 + y2 = 4 \) (Equation B)

From equations (3) and (4):
- From (3): \( x2 + x3 = 0 \) (Equation C)
- From (4): \( y2 + y3 = -2 \) (Equation D)

From equations (5) and (6):
- From (5): \( x1 + x3 = 4 \) (Equation E)
- From (6): \( y1 + y3 = -2 \) (Equation F)

### Step 4: Express Variables
From Equation C:
\[ x3 = -x2 \quad \text{(7)} \]

Substituting (7) into Equation E:
\[ x1 - x2 = 4 \quad \text{(8)} \]

Now we have two equations (A and 8) involving \( x1 \) and \( x2 \):
1. \( x1 + x2 = 2 \) (Equation A)
2. \( x1 - x2 = 4 \) (Equation 8)

Adding these two equations:
\[ 2x1 = 6 \implies x1 = 3 \]

Substituting \( x1 = 3 \) into Equation A:
\[ 3 + x2 = 2 \implies x2 = -1 \]

Using (7) to find \( x3 \):
\[ x3 = -(-1) = 1 \]

### Step 5: Solve for y-coordinates
Using Equations B and D:
1. \( y1 + y2 = 4 \) (Equation B)
2. \( y2 + y3 = -2 \) (Equation D)

From Equation D:
\[ y3 = -2 - y2 \quad \text{(9)} \]

Substituting (9) into Equation F:
\[ y1 + (-2 - y2) = -2 \implies y1 - y2 = 0 \implies y1 = y2 \]

Substituting \( y1 = y2 \) into Equation B:
\[ 2y1 = 4 \implies y1 = 2 \quad \text{and thus } y2 = 2 \]

Using (9) to find \( y3 \):
\[ y3 = -2 - 2 = -4 \]

### Step 6: Final Coordinates
Thus, the coordinates of the vertices of the triangle are:
- Vertex A = (3, 2)
- Vertex B = (-1, 2)
- Vertex C = (1, -4)

### Summary of the Solution
The coordinates of the vertices of the triangle are:
- A(3, 2)
- B(-1, 2)
- C(1, -4)