Question 199372
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Calculate the slopes of the four lines containing the four sides of the quadrilateral using:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m = \frac{y_1 - y_2}{x_1 - x_2} ]


where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are pairs of the given quadrilateral vertices representing the endpoints of the side segments.


Use:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1 \parallel L_2 \ \ \Leftrightarrow\ \ m_1 = m_2]


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:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1 \perp L_2 \ \ \Leftrightarrow\ \ m_1 = -\frac{1}{m_2} \text{ and } m_1, m_2 \neq 0]


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:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d = sqrt{(x_1 - x_2)^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:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m = \frac{y_1 - y_2}{x_1 - x_2} ]


on the opposing vertices, namely DF and EG.  Then test for perpendicularity using:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1 \perp L_2 \ \ \Leftrightarrow\ \ m_1 = -\frac{1}{m_2} \text{ and } m_1, m_2 \neq 0]


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:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y - y_1 = m(x - x_1) ]


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:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}] and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}]


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
*[tex \LARGE e^{i\pi} + 1 = 0]
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