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# The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices. Q.4

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What is the best solution for the coordinate geometry questions , find the best and simple way to solve the coordinate geometry questions . The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices.

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1. 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 =Â âˆš[(3 + 1)2Â + (2 â€“ 2)2] = 4

Coordinates of O can be calculated as follows:

x = (3 â€“ 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 = 2âˆš2

Hence, each side of square = 2âˆš2

Step 3: Find coordinates of point D

Equate length measure of AD and CD

Say, if coordinate of D are (x1, y1)

AD =Â âˆš[(x1Â + 1)2Â + (y1Â â€“ 2)2]

Squaring both sides,

AD2Â = (x1Â + 1)2Â + (y1Â â€“ 2)2

Similarly, CD2Â =Â (x1Â â€“ 3)2Â + (y1Â â€“ 2)2

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

(x1Â + 1)2Â + (y1Â â€“ 2)2Â =Â (x1Â â€“ 3)2Â + (y1Â â€“ 2)2

x12Â + 1 + 2x1Â = x12Â + 9 â€“ 6x1

8x1Â = 8

x1Â = 1

Value of y1Â can be calculated as follows by using the value of x.

From step 2: each side of square = 2âˆš2

CD2Â =Â (x1Â â€“ 3)2Â + (y1Â â€“ 2)2

8 =Â (1Â â€“ 3)2Â + (y1Â â€“ 2)2

8 = 4 + (y1Â â€“ 2)2

y1Â â€“ 2 = 2

y1Â = 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 = (x2Â + 1)/2

x2Â = 1

And 2 = (y2Â + 4)/2

=> y2Â = 0

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

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