Question

assume that the units shown in the grid are in feet. (a) determine the exact length and width of the rectangle shown. (b) determine the perimeter and area. write your answer in simplest form.

Explanation:

Step1: Identify the vertices of the rectangle

Let the vertices of the rectangle be \(A(0,3)\), \(B(4, - 2)\), \(C(5,-1)\), \(D(1,4)\). Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the length and width.

Step2: Find the length

Let's find the length by calculating the distance between two non - adjacent vertices. For example, between \(A(0,3)\) and \(B(4,-2)\).
\[

$$\begin{align*} d_{AB}&=\sqrt{(4 - 0)^2+(-2 - 3)^2}\\ &=\sqrt{4^2+(-5)^2}\\ &=\sqrt{16 + 25}\\ &=\sqrt{41} \end{align*}$$

\]

Step3: Find the width

Let's find the distance between \(A(0,3)\) and \(D(1,4)\)
\[

$$\begin{align*} d_{AD}&=\sqrt{(1 - 0)^2+(4 - 3)^2}\\ &=\sqrt{1^2+1^2}\\ &=\sqrt{1+1}\\ &=\sqrt{2} \end{align*}$$

\]

Step4: Calculate the perimeter

The perimeter \(P\) of a rectangle is \(P = 2(l + w)\), where \(l=\sqrt{41}\) and \(w = \sqrt{2}\).
\[

$$\begin{align*} P&=2(\sqrt{41}+\sqrt{2})\\ &=2\sqrt{41}+2\sqrt{2} \end{align*}$$

\]

Step5: Calculate the area

The area \(A\) of a rectangle is \(A=l\times w\), so \(A=\sqrt{41}\times\sqrt{2}=\sqrt{82}\)

Answer:

(a) Length: \(\sqrt{41}\) feet, Width: \(\sqrt{2}\) feet
(b) Perimeter: \((2\sqrt{41}+2\sqrt{2})\) feet, Area: \(\sqrt{82}\) square - feet