Question

consider parallelogram ( wxyz ) below.
note that ( wxyz ) has vertices ( w(-6,-2), x(-1,6), y(7,1) ), and ( z(2,-7) ).
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.
(a) find the length of ( overline{wx} ) and the length of a side adjacent to ( overline{wx} ).
give exact answers (not decimal approximations).
length of ( overline{wx} ): ( sqrt{89} )
length of side adjacent to ( overline{wx} ): ( sqrt{89} )
(b) find the slope of ( overline{wx} ) and the slope of a side adjacent to ( overline{wx} ).
slope of ( overline{wx} ): ( \frac{8}{5} )
slope of side adjacent to ( overline{wx} ): ( -\frac{5}{8} )
(c) from parts (a) and (b), what can we conclude about parallelogram ( wxyz )? check all that apply.
( square wxyz ) is a rectangle.
( square wxyz ) is a rhombus.
( square wxyz ) is a square.
( square wxyz ) is none of these.

Explanation:

Step1: Calculate the length of \( \overline{WX} \)

Use the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
For \( W(-6,-2) \) and \( X(-1,6) \), \( x_1=-6,y_1 = - 2,x_2=-1,y_2 = 6 \).

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Step2: Calculate the length of \( \overline{XY} \)

For \( X(-1,6) \) and \( Y(7,1) \), \( x_1=-1,y_1 = 6,x_2=7,y_2 = 1 \).

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Step3: Calculate the slope of \( \overline{WX} \)

Use the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
For \( W(-6,-2) \) and \( X(-1,6) \), \( m_{WX}=\frac{6+2}{-1 + 6}=\frac{8}{5} \).

Step4: Calculate the slope of \( \overline{XY} \)

For \( X(-1,6) \) and \( Y(7,1) \), \( m_{XY}=\frac{1 - 6}{7+1}=\frac{-5}{8} \).

Step5: Analyze the properties

Since \( |WX|=|XY|=\sqrt{89} \), adjacent sides are equal (property of a rhombus).
Also, \( m_{WX}\times m_{XY}=\frac{8}{5}\times(-\frac{5}{8})=-1 \), adjacent sides are perpendicular (property of a rectangle).
A figure that is both a rhombus and a rectangle is a square.

Answer:

A. \( WXYZ \) is a rectangle.
B. \( WXYZ \) is a rhombus.
C. \( WXYZ \) is a square.