Question 101129: A rectangle is drawn on a coordinate plane so that three vertices are located at these coordinates: (-5, 2), (5, 2), and (-5, -3) What are the coordinates of the fourth vertex of the rectangle?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Be careful of this problem. There are several answers. The easiest way to see what is going on
is to make a coordinate system and plot the given points as follows:
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Plot the point (-5,+2) and label it as A
Plot the point (+5,+2) and label it as B
Plot the point (-5,-3) and label it as C
.
Now plot the point (+5,-3) and label it as D. Can you see that the quadrilateral ABDC is a
rectangle and a rectangle is a parallelogram? To convince yourself that it is, first note
that opposite sides AB and DC are both 10 units in length. Then note that opposite
sides AC and BD are both 5 units in length. Then note that lines AB and DC are both
horizontal (no change in y values along either line). Then note that lines AC and BD
are both vertical (no change in x values along either of these lines). Because opposite lines
are either horizontal or vertical, the opposite lines are parallel. With opposite sides
parallel and of the same length you have a parallelogram.
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Next, plot the point (+5,+7) and label it as E. This time can you see that quadrilateral
ACBE is a parallelogram? To convince yourself of this, first find out if the opposite
pairs of sides (AE and BC is one pair and AC and BE is the other pair) are the same length
for each pair. Then convince yourself that they are parallel by comparing the change in
the values of x and y for each pair of lines. Each line pair should have the same change
in value for x and y. For example: line AC ... Point A has an x value of -5 and point C
has an x value of -5. There is no change in x for this line. Its opposite is line EB.
Point E has an x value of +5 and Point B has an x value of +5. Therefore, it also has
no change in x values. So line EB and AC are parallel because the x values do not change
for either line. Then for lines AE and BC you can convince yourself that the x values
for both lines change by the same amount, and then that the y values for both lines
change by the same amount. Therefore, these two lines have the same slope and are therefore
parallel lines. And when two sets of parallel lines (set AE and BC and set AC and EB) intersect
a parallelogram ACBE is formed.
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Then plot the point (-15, -3) and label it as F. Then look at the quadrilateral ABCF. Is it
a parallelogram? You can do the same analysis as above. Convince yourself that lines
AB and CF are both horizontal lines (no change in y values) and are therefore parallel.
Next convince yourself that lines AF and BC have the same slope (the changes in the x values
for each line are equal, and also the change in the y values are equal). If they are the same
slope this set of lines is also parallel. So you have two sets of parallel lines intersecting
to form parallelogram ABCF.
.
Hope this isn't too confusing and you can see your way through it to understand how you can
come up with three answers.
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