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

is a square

Explanation:

AC is a horizontal line of length 4 and midpoint (2, 4).

BD is a vertical line of length 4 and midpoint (2, 4).

The diagonals are the same length, have the same midpoint, and cross at right angles. The parallelogram is a square.

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The diagonals of a rectangle are the same length and have the same midpoint. The diagonals of a rhombus have the same midpoint and cross at right angles. A rectangle that is a rhombus is a square.

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The midpoint of a segment with end coordinates (x1, y1) and (x2, y2) is ...

midpoint = ((x1 +x2)/2, (y1 +y2)/2)

The length of a horizontal or vertical line segment is the difference of the coordinates that are different. For AC, it is 4-0 = 4; for BD, it is 6-2 = 4.