Question 978962: Math Question:
Prove using coordinate geometry:
If you connect successive midpoints of a rectangle, the resulting figure is a rhombus.
(Our teacher isn't the type to explain things at all in class so I have no idea what/how to solve this)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
I call bullshit on the "teacher isn't the type..." nonsense. Your teacher gets paid a salary, said funds either derived from the tuition you pay to go to school or the property taxes your parents pay to run the public schools. Your teacher therefore has a fiduciary responsibility to "explain things". So cut the whiny "poor me" crap and use your energy to learn mathematics.
Be that as it may, on a coordinate system, create an arbitrary rectangle of dimensions  by  . Since coordinates can be translated and rotated arbitrarily, you can position the rectangle with one vertex at the origin and the sides parallel to the coordinate axes without loss of generality. Hence, your vertices are (given is the horizontal dimension), , (a,0), (a,b), (0,b)) . Using the midpoint formulas:
 and
calculate the coordinates of each of the midpoints of the four sides of the rectangle, in terms of and . For example, the side between the points ) and ) has a midpoint ) .
Then, using the distance formula:
^2\ +\ (y_1\ -\ y_2)^2})
calculate the distance between each pair of adjacent side midpoints. If the inscribed figure is a rhombus, all of the sides must be equal.
John
My calculator said it, I believe it, that settles it
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