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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:
`m_(AD) = -1/(m_(CD))`
Hence, using the properties of a square, you may find the coordinates of the vertices B and D.
If ABCD is a squre then every leg length is equal.
Also in a square all angles are equal and each angle is 90 degrees.
So if A and C are given or not given B and D will be 90 degrees. Further more the diagonals AC and BD are also equal.
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