Question

rectangle has vertices at these coordinates. (0, 8), (5, 8), (5, 0) What are the coordinates of the fourth vertex of the rectangle?

Answer 1

Answer:

  (0, 0)

Step-by-step explanation:

The given vertices lie on the horizontal line y=8 and the vertical line x=5. The vertices that do not lie on those lines have y=0 and x=0. These are the coordinates of the missing vertex: (x, y) = (0, 0)

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Alternate solution

The diagonals of a rectangle have the same midpoint, so for rectangle ABCD, we have ...

  midpoint(AC) = midpoint(BD)

The midpoint of a line segment is the average of its endpoints, so this translates to ...

  (A+C)/2 = (B+D)/2

Multiplying by 2 and subtracting B gives ...

  A + C = B + D

  A + C - B = D

So, we can find the missing vertex by adding the coordinates of the diagonal and subtracting the third vertex:

  (0, 8) + (5, 0) - (5, 8) = (0+5-5, 8+0-8) = (0, 0)

The missing vertex is (0, 0).

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Please note that this method works even if the rectangle's sides are not aligned with the axes.