# ABCD is a square A and C are given how to find B and D? Notice that the problem does not specify what is needed to be known about B and D and, excluding the fact that the internal angles B and D are required, you may suppose that  the problem provides two vertices, you need to find the coordinates of the next two vertices using the following...

Notice that the problem does not specify what is needed to be known about B and D and, excluding the fact that the internal angles B and D are required, you may suppose that  the problem provides two vertices, you need to find the coordinates of the next two vertices using the following properties such that:

- the adjacent sides are perpendicular

- the point of intersection of diagonals of the square represents the midpoint of each diagonal

Using the first property, you may write the slope of the lines AB and BC and you may use the fact that the product of slopes of orthogonal lines gives -1.

`m_(AB) = (y_B - y_A)/(x_B - x_A)`

`m_(BC) = (y_B - y_C)/(x_B - x_C)`

`m_(AB) = -1/(m_(BC))`

`(y_B - y_A)/(x_B - x_A) = -(x_B - x_C)/(y_B - y_C)`

Since the problem provides the coordinates of the vertices A and C,  hence, the only  unknowns in the equation above are the coordinates `x_B, y_B` .

Using the next property, you may write the coordinates of the midpoint O such that:

`x_O = (x_A + x_C)/2 ; y_O = (y_A + y_C)/2`

Since O represents the midpoint of BD yields:

`x_O = (x_B + x_D)/2 ; y_O = (y_B + y_D)/2`

You need to find an equation that comprises the coordinates of D using the fact that the lines AD and CD are orthogonal such that:

Hence, using the properties of  a square,  you may find the coordinates of the vertices B and D.

Approved by eNotes Editorial Team If ABCD is a squre then every leg length is equal.