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Prove that the points (-2,-1),(1,0),(4,3...

Prove that the points `(-2,-1),(1,0),(4,3),` and (1,2) are the vertices of a parallelogram. Is it a rectangle?

Text Solution

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Given points are in cyclic order.
Let `A-=(-2,-1),B-=(1,0),C-=(4,3)` and `D-=(1,2)`.
Then the coordinates of the midpoint of AC are ` ((-2+4)/(2),(-1+3)/(2))-=(1,1)`
The coordinates of the midpoint of BD are
` ((1+1)/(2),(0+2)/(2))-=(1,1)`
Thus, AC and BD have the same mdipoint. Hence, ABCD is a parallelogram. Now, we shall see whether ABCD is a rectangle or not. we have
`AC=sqrt({4-(-2)}^2+{3-(-1)}^2)=2sqrt(13)`
and `BD=sqrt((1-1)^2+(0-2)^2)=2` Clearly, `ACneBD`. So, ABCD is not a rectangle.
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