Question 466331
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Since vertices A and D have equal y-coordinates we can conclude that segment AD lies in a horizontal line.  Likewise, since vertices B and C have equal y-coordinates, segment BC lies in a horizontal line.  Since the y-coordinate of A is different than the y-coordinate of B, segments AD and BC lie in different horizontal lines and are therefore parallel to each other.


Using the coordinates of points A and B, we can calculate the slope of the line containing segment AB.  Similarly we can calculate the slope of the line containing segment CD.  I leave it as an exercise for the student to show that the two described slopes are equal and therefore the two described segments are parallel.


The slope of a horizontal line is zero, therefore the slope of a vertical line is undefined.  One pair of sides is horizontal and the other pair has been shown to have a defined slope.  Therefore, we have disabused ourselves of the notion that the angles at each of the vertices might possibly be right angles, and thereby have excluded <i><b>rectangle</b></i> as an appropriate description for ABCD.


The measure of AD can clearly be seen to be 9 (square root of -9 squared plus 0 squared).


The measure of AB is *[tex \Large \sqrt{(-2)^2\ +\ (4)^2].  I'll leave it to you to verifiy that this is not equal to 9, thereby excluding <i><b>rhombus</b></i> as a possible description of ABCD.


That leaves <i><b>quadrilateral</b></i> or <i><b>parallelogram</b></i>. Since we have shown that there are two pairs of parallel sides, the answer should be obvious at this point.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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