Question

graph the quadrilateral on graph paper, then answer the questions.
7 the vertices of quadrilateral pqrs are: ( p(3,8) ) ( q(7,6) ) ( r(1,-6) ) ( s(-3,-4) )
(a) complete the steps below to prove whether quadrilateral pqrs is a rectangle.
what are the slopes of each side?
( pq ) is
( rs ) is
( qr ) is
( ps ) is
(b) prove whether quadrilateral pqrs is a rectangle. use your answers from the first part along with any other additional information to support your answer.
remember you must type the slopes here in your paragraph answer.

Explanation:

Step1: Calculate slope of \(PQ\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For \(P(3,8)\) and \(Q(7,6)\), \(m_{PQ}=\frac{6 - 8}{7 - 3}=\frac{-2}{4}=-\frac{1}{2}\)

Step2: Calculate slope of \(RS\)

For \(R(1,-6)\) and \(S(-3,-4)\), \(m_{RS}=\frac{-4-(-6)}{-3 - 1}=\frac{2}{-4}=-\frac{1}{2}\)

Step3: Calculate slope of \(QR\)

For \(Q(7,6)\) and \(R(1,-6)\), \(m_{QR}=\frac{-6 - 6}{1 - 7}=\frac{-12}{-6}=2\)

Step4: Calculate slope of \(PS\)

For \(P(3,8)\) and \(S(-3,-4)\), \(m_{PS}=\frac{-4 - 8}{-3 - 3}=\frac{-12}{-6}=2\)

Step5: Check for rectangle properties

A rectangle has opposite sides parallel (equal slopes) and adjacent sides perpendicular (product of slopes = - 1).
\(m_{PQ}=m_{RS}=-\frac{1}{2}\), \(m_{QR}=m_{PS}=2\). Also, \(m_{PQ}\times m_{QR}=-\frac{1}{2}\times2=-1\), \(m_{QR}\times m_{RS}=2\times(-\frac{1}{2})=-1\), \(m_{RS}\times m_{PS}=-\frac{1}{2}\times2=-1\), \(m_{PS}\times m_{PQ}=2\times(-\frac{1}{2})=-1\)

Answer:

(a) \(PQ:-\frac{1}{2}\), \(RS:-\frac{1}{2}\), \(QR:2\), \(PS:2\)
(b) Quadrilateral \(PQRS\) is a rectangle. Since \(m_{PQ}=m_{RS}\) and \(m_{QR}=m_{PS}\) (opposite sides are parallel) and \(m_{PQ}\times m_{QR}=- 1\), \(m_{QR}\times m_{RS}=-1\), \(m_{RS}\times m_{PS}=-1\), \(m_{PS}\times m_{PQ}=-1\) (adjacent sides are perpendicular)