Question

consider parallelogram pqrs below.
note that pqrs has vertices p(-6, 6), q(1, 1), r(6, -6), and s(-1, -1).
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.
(a) find the slope of \\( \overline { p s } \\) and the slope of a side adjacent to \\( \overline { p s } \\).
slope of \\( \overline { p s } \\):
slope of side adjacent to \\( \overline { p s } \\):
(b) find the length of \\( \overline { p s } \\) and the length of a side adjacent to \\( \overline { p s } \\).
give exact answers (not decimal approximations).
length of \\( \overline { p s } \\):
length of side adjacent to \\( \overline { p s } \\):
(c) from parts (a) and (b), what can we conclude about parallelogram pqrs? check all that apply.
\\( \square pqrs \\) is a rectangle.
\\( \square pqrs \\) is a rhombus.
\\( \square pqrs \\) is a square.
\\( \square pqrs \\) is none of these.

Explanation:

Step1: Calculate the slope of \(\overline{PS}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(-6,6)\) and \(S(-1,-1)\), \(m_{PS}=\frac{-1 - 6}{-1-(-6)}=\frac{-7}{5}\)

Step2: Calculate the slope of a side adjacent to \(\overline{PS}\)

Let's take side \(\overline{PQ}\). For points \(P(-6,6)\) and \(Q(1,1)\), \(m_{PQ}=\frac{1 - 6}{1-(-6)}=\frac{-5}{7}\)

Step3: Calculate the length of \(\overline{PS}\)

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(P(-6,6)\) and \(S(-1,-1)\), \(d_{PS}=\sqrt{(-1 + 6)^2+(-1 - 6)^2}=\sqrt{25 + 49}=\sqrt{74}\)

Step4: Calculate the length of a side adjacent to \(\overline{PS}\)

For side \(\overline{PQ}\) with \(P(-6,6)\) and \(Q(1,1)\), \(d_{PQ}=\sqrt{(1 + 6)^2+(1 - 6)^2}=\sqrt{49+25}=\sqrt{74}\)

Answer:

(a) Slope of \(\overline{PS}\): \(\frac{-7}{5}\), Slope of side adjacent to \(\overline{PS}\): \(\frac{-5}{7}\)
(b) Length of \(\overline{PS}\): \(\sqrt{74}\), Length of side adjacent to \(\overline{PS}\): \(\sqrt{74}\)
(c) Since the product of slopes \(m_{PS}\times m_{PQ}=\frac{-7}{5}\times\frac{-5}{7} = 1
eq - 1\) (so not a rectangle), and \(|\overline{PS}|=|\overline{PQ}|=\sqrt{74}\). A rhombus has all sides equal. A square requires all sides equal and adjacent sides perpendicular (slopes product \(-1\)). So \(PQRS\) is a rhombus.
So for part (c): \(PQRS\) is a rhombus.