Let ABCD is a square, where A(-1,2) and B(3,2). And Point O is the point of intersection of AC and BD

To Find: Coordinate of points B and D.

Step 1: Find distance between A and C and coordinates of point O.

We know that, diagonals of a square are equal and bisect each other.

AC = √[(3 + 1)² + (2 – 2)²] = 4

Coordinates of O can be calculated as follows:

x = (3 – 1)/2 = 1 and y = (2 + 2)/2 = 2

So, O(1,2)

Step 2: Find the side of the square using Pythagoras theorem

Let a be the side of square and AC = 4

From right triangle, ACD,

a = 2√2

Hence, each side of square = 2√2

Step 3: Find coordinates of point D

Equate length measure of AD and CD

Say, if coordinate of D are (x₁, y₁)

AD = √[(x₁ + 1)² + (y₁ – 2)²]

Squaring both sides,

AD² = (x₁ + 1)² + (y₁ – 2)²

Similarly, CD² = (x₁ – 3)² + (y₁ – 2)²

Since all sides of a square are equal, which means AD = CD

(x₁ + 1)² + (y₁ – 2)² = (x₁ – 3)² + (y₁ – 2)²

x₁² + 1 + 2x₁ = x₁² + 9 – 6x₁

8x₁ = 8

x₁ = 1

Value of y₁ can be calculated as follows by using the value of x.

From step 2: each side of square = 2√2

CD² = (x₁ – 3)² + (y₁ – 2)²

8 = (1 – 3)² + (y₁ – 2)²

8 = 4 + (y₁ – 2)²

y₁ – 2 = 2

y₁ = 4

Hence, D = (1, 4)

Step 4: Find coordinates of point B

From line segment, BOD

Coordinates of B can be calculated using coordinates of O; as follows:

Earlier, we had calculated O = (1, 2)

Say B = (x₂, y₂)

For BD;

1 = (x₂ + 1)/2

x₂ = 1

And 2 = (y₂ + 4)/2

=> y₂ = 0

Therefore, the coordinates of required points are B = (1,0) and D = (1,4)