If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.

#### Solution

The distance *d* between two points `(x_1,y_1)` and `(x_2,y_2)` is given by `the formula

`d= sqrt((x_1- x_2)^2+(y_1 -y_2)^2)`

In a square, all the sides are of equal length. The diagonals are also equal to each other. Also in a square, the diagonal is equal to `sqrt2` times the side of the square.

Here let the two points which are said to be the opposite vertices of a diagonal of a square be *A*(5*,*4)* *and *C*(1*,**−*6).

Let us find the distance between them which is the length of the diagonal of the square.

`AC = sqrt((5 - 1)^2 + (4 = 6)^2)`

`=sqrt((4)^2 + (10)^2)`

`= sqrt(16 + 100)`

`AC = 2sqrt29`

Now we know that in a square,

Side of the square = `"Diagonal of the square"/sqrt2`

Substituting the value of the diagonal we found out earlier in this equation we have,

Side of the square = `(2sqrt29)/sqrt2`

side of the square = `sqrt58`

Now, a vertex of a square has to be at equal distances from each of its adjacent vertices.

Let P(x, y) represent another vertex of the same square adjacent to both ‘*A*’ and ‘*C*

`AP = sqrt((5 - x)^2 + (4 - y)^2)`

`CP = sqrt((1 - x)^2 + (-6-y)^2)`

But these two are nothing but the sides of the square and need to be equal to each other.

AP = CP

`sqrt((5 - x)^2 + (4 - y)) = sqrt((1 - x)^2 + (-6 - y)^2)`

Squaring on both sides we have,

`(5 -x)^2 + (4 - y)^2 = (1 - x)^2 + (-6 - y)^2`

`25 + x^2 - 10x + 16 + y^2 - 8y = 1 + x^2 - 2x + 36 + y^2 + 12y`

8x + 20y = 4

2x + 5y = 1

From this we have, x = `(1- 5y)/2`

Substituting this value of ‘*x*’ and the length of the side in the equation for ‘*AP*’ we have,

`AP = sqrt((5 - x)^2 + (4 - y)^2)`

`sqrt(58) = sqrt((5 - x)^2 + (4 - y)^2)`

Squaring on both sides,

`58 = (5 - x)^2 + (4 - y)^2`

`58 = (5 - ((1 - 5y)/2))^2 + (4 - y)^2`

`58 = ((9 + 5y)/2)^2 + (4 - y)^2`

`58 = (81 + 25y^2 + +90y)/4 + 16 + y^2 - 8y`

`232 = 81 + 25y^2 + 90y + 64 + 4y^2 - 32y`

`87 = 29y^2 + 58y`

We have a quadratic equation. Solving for the roots of the equation we have,

`29y^2 + 58y - 87 = 0`

`29y^2 + 87y - 29y - 87 = 0`

29y(y + 3) - 29(y + 3) = 0

(y + 3)(29y - 29) = 0

(y + 3)(y - 1) = 0

The roots of this equation are −3 and 1.

Now we can find the respective values of ‘*x*’ by substituting the two values of ‘*y*’

When y = -3

`x = (1 - 5(-3))/2`

`= (1 + 15)/2`

x= 8

when y = 1

`x = (1- 5(1))/2`

`= (1-5)/2`

x = -2

Therefore the other two vertices of the square are (8, -3) and (-2, 1).