A(-2,4),C(4,10) and D(-2,10) are the vertices of a square ABCD. Use the graphical method to find the co-ordinates of the fouth vertex B. Find :
(i) the co-ordinates of the mid point of BC
(ii) the co-ordinates of the mix point of CD and
(iii) the co-ordinates of the point of intersection of the diagonals of the square ABCD.

Answer

To find the coordinates of the fourth vertex B of square ABCD, where A(-2, 4), C(4, 10), and D(-2, 10) are given, we can follow these steps:

### Step 1: Plot the Given Points
We start by plotting the points A, C, and D on a Cartesian plane.

- A(-2, 4)
- C(4, 10)
- D(-2, 10)

### Step 2: Identify the Coordinates of Vertex B
Since ABCD is a square, we know that:
- The sides are equal in length.
- The angles between the sides are 90 degrees.

From the points plotted:
- A is at (-2, 4).
- D is directly above A at (-2, 10).
- C is to the right of D at (4, 10).

To find B, we note that it must be directly above A and directly to the left of C. Therefore, the coordinates of B will be:
- B(4, 4)

### Step 3: Find the Midpoint of BC
The coordinates of B are (4, 4) and C are (4, 10). The midpoint (M) of line segment BC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of B and C:
\[ M_{BC} = \left( \frac{4 + 4}{2}, \frac{4 + 10}{2} \right) = \left( 4, \frac{14}{2} \right) = (4, 7) \]

### Step 4: Find the Midpoint of CD
The coordinates of C are (4, 10) and D are (-2, 10). The midpoint (N) of line segment CD can be calculated similarly:
\[ N = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of C and D:
\[ M_{CD} = \left( \frac{4 + (-2)}{2}, \frac{10 + 10}{2} \right) = \left( \frac{2}{2}, \frac{20}{2} \right) = (1, 10) \]

### Step 5: Find the Point of Intersection of the Diagonals
The diagonals of square ABCD are AC and BD. The coordinates of A are (-2, 4) and C are (4, 10). The midpoint of diagonal AC can be calculated as:
\[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of A and C:
\[ P_{AC} = \left( \frac{-2 + 4}{2}, \frac{4 + 10}{2} \right) = \left( \frac{2}{2}, \frac{14}{2} \right) = (1, 7) \]

Since the diagonals of a square bisect each other, the point of intersection of the diagonals is also the midpoint of diagonal BD, which is also (1, 7).

### Summary of Results
1. Coordinates of vertex B: (4, 4)
2. Midpoint of BC: (4, 7)
3. Midpoint of CD: (1, 10)
4. Point of intersection of diagonals: (1, 7)