Question 199372: Analytic Geometry
14. Use analytic geometry to classify the quadrilateral with vertices D(10,0), E(2,4), F(-8,-6), and G(6,-8). Explain your reasoning and show all your work.
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Answer by solver91311(24713)   (Show Source):
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Calculate the slopes of the four lines containing the four sides of the quadrilateral using:
where ) and ) are pairs of the given quadrilateral vertices representing the endpoints of the side segments.
Use:
to determine if any pairs of sides are parallel to each other.
If you have two sets of parallel sides, then you have some form of parallelogram, either a general parallelogram, a rhombus, a rectangle or a square.
If you do have a parallelogram, then use:
to determine if the adjacent sides are mutually perpendicular. If so, then you have a rectangle or a square, if not, a general parallelogram or rhombus.
At this point you can use the distance formula to determine if two adjacent sides have the same length:
^2 + (y_1 - y_2)^2})
If they do, you have a square or a rhombus based on the perpendicularity determination made above; if not, it is a rectangle or general parallelogram -- again based on perpendicularity.
If you only have one pair of parallel sides, then you have a trapezoid (trapezium if you are British).
If you have no pairs of parallel sides, you either have a kite or a general quadrilateral. A kite has the property that the diagonals are perpendicular. Use:
on the opposing vertices, namely DF and EG. Then test for perpendicularity using:
If the diagonals are not perpendicular, you have a general quadrilateral. If they are perpendicular, you have one additional test to perform. In a kite, one of the diagonals bisects the other (but not vice versa).
Use the point-slope form of the equation of a line:
 )
to write an equation for the lines containing the diagonal segments.
Solve the system of two equations to determine their point of intersection. Then use:
 and
to determine the mid-points of each of the diagonal segments. If one or the other mid-point is the same as the point of intersection of the diagonals, then you have a kite, otherwise it is a general quadrilateral.
John
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