Question

consider parallelogram pqrs below.
note that pqrs has vertices ( p(-3, 2), q(4, 4), r(6, -3) ), and ( s(-1, -5) ).
answer the following to determine if the parallelogram is a recta rhombus, square, or none of these.
(a) find the length of ( overline{pq} ) and the length of a side adjacent to ( overline{pq} ).
give exact answers (not decimal approximations).
length of ( overline{pq} ):
length of side adjacent to ( overline{pq} ):
(b) find the slope of ( overline{pq} ) and the slope of a side adjacent to ( overline{pq} ).
slope of ( overline{pq} ):
slope of side adjacent to ( overline{pq} ):
(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 length of $\overline{PQ}$

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For $P(-3,2)$ and $Q(4,4)$:
$$\overline{PQ}=\sqrt{(4-(-3))^2+(4-2)^2}=\sqrt{7^2+2^2}=\sqrt{49+4}=\sqrt{53}$$

Step2: Calculate length of adjacent side $\overline{PS}$

Use distance formula for $P(-3,2)$ and $S(-1,-5)$:
$$\overline{PS}=\sqrt{(-1-(-3))^2+(-5-2)^2}=\sqrt{2^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}$$

Step3: Calculate slope of $\overline{PQ}$

Use slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$$m_{PQ}=\frac{4-2}{4-(-3)}=\frac{2}{7}$$

Step4: Calculate slope of adjacent $\overline{PS}$

Use slope formula for $P(-3,2)$ and $S(-1,-5)$:
$$m_{PS}=\frac{-5-2}{-1-(-3)}=\frac{-7}{2}$$

Step5: Analyze properties for classification

1. Adjacent sides are equal ($\sqrt{53}=\sqrt{53}$), so it is a rhombus.
2. Check if slopes are perpendicular: $m_{PQ} \times m_{PS} = \frac{2}{7} \times \frac{-7}{2} = -1$, so adjacent sides are perpendicular. A rhombus with perpendicular sides is a square, and all squares are rectangles.

Answer:

(a)

Length of $\overline{PQ}$: $\boldsymbol{\sqrt{53}}$
Length of side adjacent to $\overline{PQ}$: $\boldsymbol{\sqrt{53}}$

(b)

Slope of $\overline{PQ}$: $\boldsymbol{\frac{2}{7}}$
Slope of side adjacent to $\overline{PQ}$: $\boldsymbol{-\frac{7}{2}}$

(c)

• $\boldsymbol{P Q R S}$ is a rectangle.
• $\boldsymbol{P Q R S}$ is a rhombus.
• $\boldsymbol{P Q R S}$ is a square.