SOLUTION: if you put these points on a graph (-5,-2), (5,0), (6,-5), (-4,-7) is it a rhombus, parallelogram, rectangle, or square

Algebra ->  Parallelograms -> SOLUTION: if you put these points on a graph (-5,-2), (5,0), (6,-5), (-4,-7) is it a rhombus, parallelogram, rectangle, or square      Log On


   

Question 410622: if you put these points on a graph (-5,-2), (5,0), (6,-5), (-4,-7) is it a rhombus, parallelogram, rectangle, or square
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
if you put these points on a graph (-5,-2), (5,0), (6,-5), (-4,-7) is it a rhombus, parallelogram, rectangle, or square:
 
Hmm!  It it were a square it would also be a rhombus, 
a rectangle, and a parallelogram.

If it were a rhombus it would also be a parallelogram.

If it were a rectangle it would also be a parallelogram.

So we plot the points and connect them:

(-5,-2), (5,0), (6,-5), (-4,-7)



It looks like a rectangle and a parallelogram, but not a square or rhombus.

To show that it is a parallelogram, we show that both pairs of opposite
sides are parallel.  To do that we find the slope of the sides, and 
observe that they are the same.

The formula for the slope is m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29


Finding the slopes of the two long sides:

A. The slope of the upper long side:

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29
m=%280-%28-2%29%29%2F%285-%28-5%29%29
m=%280%2B2%29%2F%285%2B5%29
m=2%2F10
m=1%2F5

B. The slope of the lower long side:

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29
m=%28-5-%28-7%29%29%2F%286-%28-4%29%29
m=%28-5%2B7%29%2F%286%2B4%29
m=2%2F10
m=1%2F5

So the pair of longer sides are parallel.


Finding the slopes of the two shorter sides:

A. The slope of the leftmost shorter side:

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29
m=%28-7-%28-2%29%29%2F%28-4-%28-5%29%29
m=%28-7%2B2%29%2F%28-4%2B5%29
m=-5%2F1
m=-5

B. The slope of the rightmost shorter side:

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29
m=%28-5-0%29%2F%286-5%29
m=%28-5%29%2F1
m=-5

So the pair of shorter sides are parallel.

That proves the figure is a parallelogram.

Now if one of its interior angles is a right angle, then
the parallelogram is a rectangle.  We have shown that it is
because the slope of the longer sides is 1%2F5 and the
slope of the shorter sides is -5 and one is the
"negative reciprocal" of the other, (or you can say their
produce is -1), so that means they are perpendicular and
so it is a rectangle.

However we have to rule out it's being a rhombus or square,
even though it doesn't look like one.

We find the legths of two adjacent sides:

The formula for the distance between two points, or the
length of the line joining them is:

d+=+sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29

Finding the length of the upper side:

d+=+sqrt%28%285-%28-5%29%29%5E2%2B%280-%28-2%29%29%5E2%29
d+=+sqrt%28%285%2B5%29%5E2%2B%280%2B2%29%5E2%29
d+=+sqrt%2810%5E2%2B2%5E2%29
d+=+sqrt%28100%2B4%29
d+=+sqrt%28104%29
d+=+sqrt%284%2A26%29
d+=+sqrt%284%29sqrt%2826%29
d+=+2sqrt%2826%29

Finding the length of the left side:

d+=+sqrt%28%28-4-%28-5%29%29%5E2%2B%28-7-%28-2%29%29%5E2%29
d+=+sqrt%28%28-4%2B5%29%5E2%2B%28-7%2B2%29%5E2%29
d+=+sqrt%281%5E2%2B%28-5%29%5E2%29
d+=+sqrt%281%2B25%29
d+=+sqrt%2826%29

Those sides are not the same length, so the rectangle is not 
a square or a rhombus.

Therefore it is a rectangle and a parallelogram, and not a square 
and not a rhombus.

Edwin