The co-ordinates of three consecutive vertices of a parallelogram are (2, 0) , (4, 1) and (6, 4). Find the co-ordinates of its 4th vertex.

Answer

To find the coordinates of the fourth vertex of the parallelogram given three consecutive vertices A(2, 0), B(4, 1), and C(6, 4), we can follow these steps:

### Step 1: Identify the vertices
Let’s denote the vertices of the parallelogram as follows:
- A = (2, 0)
- B = (4, 1)
- C = (6, 4)
- D = (x, y) (the vertex we need to find)

### Step 2: Find the midpoint of diagonal AC
The diagonals of a parallelogram bisect each other. Therefore, we can find the midpoint O of diagonal AC. The coordinates of the midpoint O can be calculated using the midpoint formula:
\[ O = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Here, A = (2, 0) and C = (6, 4):
\[ O_x = \frac{2 + 6}{2} = \frac{8}{2} = 4 \]
\[ O_y = \frac{0 + 4}{2} = \frac{4}{2} = 2 \]
Thus, the coordinates of point O are (4, 2).

### Step 3: Use the property of the midpoint for diagonal BD
Since O is also the midpoint of diagonal BD, we can set up the midpoint formula for points B and D:
\[ O = \left(\frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}\right) \]
Substituting the coordinates of B(4, 1) and D(x, y):
\[ (4, 2) = \left(\frac{4 + x}{2}, \frac{1 + y}{2}\right) \]

### Step 4: Set up equations from the midpoint
From the x-coordinates:
\[ 4 = \frac{4 + x}{2} \]
Multiplying both sides by 2:
\[ 8 = 4 + x \implies x = 8 - 4 = 4 \]

From the y-coordinates:
\[ 2 = \frac{1 + y}{2} \]
Multiplying both sides by 2:
\[ 4 = 1 + y \implies y = 4 - 1 = 3 \]

### Step 5: Conclusion
Thus, the coordinates of the fourth vertex D are:
\[ D = (4, 3) \]