Question

1. it is given that wxyz is a parallelogram. plot the points and graph the figure. prove what type of parallelogram you have and justify your answer by using theorems about quadrilaterals. remember to show all steps.

w(7, -1), x(3, 6), y(-1, -1), z(3, -8)

Explanation:

Step1: Identify the distance formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). We will use this to find the lengths of \(WX\), \(XY\), \(YZ\), and \(ZW\).

Step2: Calculate the length of \(WX\)

Given \(W(7, - 1)\) and \(X(3,6)\).
Using the distance formula:
\(WX=\sqrt{(3 - 7)^2+(6-(-1))^2}=\sqrt{(-4)^2+(7)^2}=\sqrt{16 + 49}=\sqrt{65}\)

Step3: Calculate the length of \(XY\)

Given \(X(3,6)\) and \(Y(-1,-1)\).
Using the distance formula:
\(XY=\sqrt{(-1 - 3)^2+(-1 - 6)^2}=\sqrt{(-4)^2+(-7)^2}=\sqrt{16 + 49}=\sqrt{65}\)

Step4: Calculate the length of \(YZ\)

Given \(Y(-1,-1)\) and \(Z(3,-8)\).
Using the distance formula:
\(YZ=\sqrt{(3-(-1))^2+(-8 - (-1))^2}=\sqrt{(4)^2+(-7)^2}=\sqrt{16 + 49}=\sqrt{65}\)

Step5: Calculate the length of \(ZW\)

Given \(Z(3,-8)\) and \(W(7,-1)\).
Using the distance formula:
\(ZW=\sqrt{(7 - 3)^2+(-1-(-8))^2}=\sqrt{(4)^2+(7)^2}=\sqrt{16 + 49}=\sqrt{65}\)

Step6: Analyze the lengths

Since \(WX = XY=YZ = ZW=\sqrt{65}\), all four sides of the quadrilateral \(WXYZ\) are equal. In a parallelogram, if all four sides are equal, then the parallelogram is a rhombus. Also, we can check the slopes to confirm it's a parallelogram (opposite sides parallel). The slope of \(WX\): \(m_{WX}=\frac{6 - (-1)}{3 - 7}=\frac{7}{-4}=-\frac{7}{4}\), slope of \(YZ\): \(m_{YZ}=\frac{-8-(-1)}{3-(-1)}=\frac{-7}{4}=-\frac{7}{4}\), slope of \(XY\): \(m_{XY}=\frac{-1 - 6}{-1 - 3}=\frac{-7}{-4}=\frac{7}{4}\), slope of \(ZW\): \(m_{ZW}=\frac{-1-(-8)}{7 - 3}=\frac{7}{4}\). Opposite sides have equal slopes (parallel), and all sides are equal, so it's a rhombus.

Answer:

The quadrilateral \(WXYZ\) is a rhombus (since it is a parallelogram with all four sides equal).