Question 466331: What is he most precise name for quadrilateral ABCD with vertices A(-5,2), B(-3,6), C(6,6) and D(4,2)?
a) quadrilateral
b) rectangle
c) parallelogram
d) rhombus
Answer by solver91311(24713)   (Show Source):
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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 rectangle 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 ^2\ +\ (4)^2) . I'll leave it to you to verifiy that this is not equal to 9, thereby excluding rhombus as a possible description of ABCD.
That leaves quadrilateral or parallelogram. Since we have shown that there are two pairs of parallel sides, the answer should be obvious at this point.
John
My calculator said it, I believe it, that settles it
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