Question

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graph the quadrilateral on graph paper, then answer the questions.
the vertices of quadrilateral pqrs are:
p(4,7) q(11,5) r(5, - 16) s(-2, - 14)
(a) complete the steps below to prove whether quadrilateral pqrs is a rectangle.
what are the slopes of each side?
pq = - 2/7
rs = 7/2
qr = - 21/6
ps = 1/2
(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.

Explanation:

Step1: Find the slope of \(PQ\)

Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(P(4,7)\) and \(Q(11,5)\), \(x_1 = 4,y_1=7,x_2 = 11,y_2 = 5\).

$$m_{PQ}=\frac{5 - 7}{11 - 4}=\frac{-2}{7}$$

Step2: Find the slope of \(QR\)

For points \(Q(11,5)\) and \(R(5,-16)\), \(x_1 = 11,y_1 = 5,x_2=5,y_2=-16\).

$$m_{QR}=\frac{-16 - 5}{5 - 11}=\frac{-21}{-6}=\frac{7}{2}$$

Step3: Find the slope of \(RS\)

For points \(R(5,-16)\) and \(S(-2,-14)\), \(x_1 = 5,y_1=-16,x_2=-2,y_2=-14\).

$$m_{RS}=\frac{-14+16}{-2 - 5}=\frac{2}{-7}=-\frac{2}{7}$$

Step4: Find the slope of \(PS\)

For points \(P(4,7)\) and \(S(-2,-14)\), \(x_1 = 4,y_1 = 7,x_2=-2,y_2=-14\).

$$m_{PS}=\frac{-14 - 7}{-2 - 4}=\frac{-21}{-6}=\frac{7}{2}$$

Step5: Check if it is a rectangle

In a rectangle, opposite sides are parallel (slopes are equal) and adjacent sides are perpendicular (product of slopes is \(- 1\)).

• Opposite sides: \(m_{PQ}=m_{RS}=-\frac{2}{7}\) and \(m_{QR}=m_{PS}=\frac{7}{2}\)
• Adjacent sides: \(m_{PQ}\times m_{QR}=-\frac{2}{7}\times\frac{7}{2}=-1\), \(m_{QR}\times m_{RS}=\frac{7}{2}\times(-\frac{2}{7})=-1\), \(m_{RS}\times m_{PS}=-\frac{2}{7}\times\frac{7}{2}=-1\), \(m_{PS}\times m_{PQ}=\frac{7}{2}\times(-\frac{2}{7})=-1\)

Answer:

• \(PQ =-\frac{2}{7}\), \(QR=\frac{7}{2}\), \(RS =-\frac{2}{7}\), \(PS=\frac{7}{2}\)
• Quadrilateral \(PQRS\) is a rectangle.