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 fourth vertex B. Also, find :

(i) the co-ordinates of the mid-point of BC;

(ii) the co-ordinates of the mid-point of CD and

(iii) the co-ordinates of the point of intersection of the diagonals of the square ABCD.

Plot the points A (- 2, 4), C (4, 10) and D (- 2, 10) on the graph paper. Join point A with D and D with C.

From the graph, it is clear that the horizontal distance between the points C (4, 10) and D (-2, 10) is 6 units and the vertical distance between the points A (-2, 4) and D (-2, 10) is 6 units. Therefore, the vertical distance between the points C (4, 10) and B must be 6 units and the horizontal distance between the points A (-2, 4) and B must be 6 units.

Now, complete the square ABCD and read the coordinates of point B, as shown on the graph, B = (4, 4).

The midpoint of BC lies exactly halfway between B(4, 4) and C(4, 10). On the graph, this midpoint is at E(4, 7), as it is 3 units from both B and C.

The midpoint of CD lies exactly halfway between C(4, 10) and D(-2, 10). On the graph, this midpoint is at F(1, 10), as it is 3 units from both C and D.

The coordinates of the midpoint of diagonals of the square is G = (1, 7).

Hence, the co-ordinates of the mid-point of BC = (4, 7), the co-ordinates of the mid-point of CD = (1, 10) and the co-ordinates of the point of intersection of the diagonals of the square ABCD = (1, 7).