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 { w x } \\) and the length of a side adjacent to \\( \overline { w x } \\).
give exact answers (not decimal approximations).
length of \\( \overline { w x } \\):
length of side adjacent to \\( \overline { w x } \\):
(b) find the slope of \\( \overline { w x } \\) and the slope of a side adjacent to \\( \overline { w x } \\).
slope of \\( \overline { w x } \\):
slope of side adjacent to \\( \overline { w x } \\):
(c) from parts (a) and (b), what can we conclude about parallelogram wxyz? check all that apply.
\\( \square w x y z \\) is a rectangle.
\\( \square w x y z \\) is a rhombus.
\\( \square w x y z \\) is a square.
\\( \square w x y z \\) is none of these.

Explanation:

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

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points \(W(-6,-2)\) and \(X(-1,6)\):
\(x_1=-6,y_1 = - 2,x_2=-1,y_2 = 6\)
\(d_{WX}=\sqrt{(-1-(-6))^2+(6 - (-2))^2}=\sqrt{(5)^2+(8)^2}=\sqrt{25 + 64}=\sqrt{89}\)

Step2: Find the length of a side adjacent to \(\overline{WX}\)

Take \(\overline{XY}\). For points \(X(-1,6)\) and \(Y(7,1)\)
\(x_1=-1,y_1 = 6,x_2=7,y_2 = 1\)
\(d_{XY}=\sqrt{(7-(-1))^2+(1 - 6)^2}=\sqrt{(8)^2+(-5)^2}=\sqrt{64+25}=\sqrt{89}\)

Step3: Find 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: Find the slope of a side adjacent to \(\overline{WX}\)

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

Step5: Analyze the properties

Since \(m_{WX}\times m_{XY}=\frac{8}{5}\times\frac{-5}{8}=- 1\), adjacent sides are perpendicular. Also, \(d_{WX}=d_{XY}=\sqrt{89}\)

Answer:

(a) Length of \(\overline{WX}\): \(\sqrt{89}\); Length of side adjacent to \(\overline{WX}\): \(\sqrt{89}\)
(b) Slope of \(\overline{WX}\): \(\frac{8}{5}\); Slope of side adjacent to \(\overline{WX}\): \(-\frac{5}{8}\)
(c) \(WXYZ\) is a rectangle; \(WXYZ\) is a rhombus; \(WXYZ\) is a square