Question

consider parallelogram pqrs below.
note that pqrs has vertices p(-2, 4), q(-6, -1), r(-1, 3), and s(3, 8).
answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.
(a) find the slope of \\( \overline{qr} \\) and the slope of a side adjacent to \\( \overline{qr} \\).
slope of \\( \overline{qr} \\):
slope of side adjacent to \\( \overline{qr} \\):
(b) find the length of \\( \overline{qr} \\) and the length of a side adjacent to \\( \overline{qr} \\).
give exact answers (not decimal approximations).
length of \\( \overline{qr} \\):
length of side adjacent to \\( \overline{qr} \\):
(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 \( \overline{QR} \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \( Q(-6,-1) \) and \( R(-1,3) \), we have \( x_1=-6,y_1 = - 1,x_2=-1,y_2 = 3 \).

$$m_{QR}=\frac{3-(-1)}{-1-(-6)}=\frac{3 + 1}{-1 + 6}=\frac{4}{5}$$

Step2: Find the slope of a side adjacent to \( \overline{QR} \)

Adjacent sides to \( \overline{QR} \) are \( \overline{PQ} \) or \( \overline{RS} \). Let's use \( \overline{PQ} \) with points \( P(-2,4) \) and \( Q(-6,-1) \).

$$m_{PQ}=\frac{-1 - 4}{-6-(-2)}=\frac{-5}{-4}=\frac{5}{4}$$

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

The distance formula is \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( Q(-6,-1) \) and \( R(-1,3) \),

$$d_{QR}=\sqrt{(-1+6)^2+(3 + 1)^2}=\sqrt{5^2+4^2}=\sqrt{25 + 16}=\sqrt{41}$$

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

Using \( \overline{PQ} \) with \( P(-2,4) \) and \( Q(-6,-1) \),

$$d_{PQ}=\sqrt{(-6 + 2)^2+(-1 - 4)^2}=\sqrt{(-4)^2+(-5)^2}=\sqrt{16+25}=\sqrt{41}$$

Answer:

(a) Slope of \( \overline{QR}:\frac{4}{5} \), Slope of side adjacent to \( \overline{QR}:\frac{5}{4} \)
(b) Length of \( \overline{QR}:\sqrt{41} \), Length of side adjacent to \( \overline{QR}:\sqrt{41} \)
(c) Since the product of the slopes \( m_{QR}\times m_{PQ}=\frac{4}{5}\times\frac{5}{4} = 1
eq- 1 \) (so it's not a rectangle, and since adjacent sides are equal in length (\( d_{QR}=d_{PQ}=\sqrt{41} \)), \( PQRS \) is a rhombus.
So for part (c): \( PQRS \) is a rhombus.