Question

A parallelogram has vertices A(0, 4), B(2, 2), C(4, 4), and D(2,6). Is this parallelogram a square? Explain why or why not.

Answer 1

If we take the adjacent side AD and DC  and work out their slopes we get 
AD = (6-4)/(2-0)  = 1 and DC has slope (6-4)/(2-4)  = -1 . This shows that the angle between the 2 lines is 90 degrees so it looks like  this is a square. If you include the slopes  of each adjacent line they are at right angles too.  Now to prove its a square  each line must be of equal length and you'll find this is the case too. 

Answer 2

A square is a special parallelogram that has 4 congruent sides and 4 right angles. Using the distance formula, you can prove that all 4 sides have a length of the square root of 8. This proves the parallelogram is a rhombus. Next, find the slope of all 4 sides.  The slopes of sides AB and CD are –1, and the slopes of sides BC and DA are 1. Since adjacent sides have opposite reciprocal slopes they are perpendicular and form right angles. This proves that parallelogram ABCD is a square.