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 length and width using distance formula

The rectangle has vertices at (-2,0), (0,3), (2,0), (0, - 3). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for the length (longer side) between (0,3) and (0, - 3), \(x_1=x_2 = 0\), \(y_1 = 3\), \(y_2=-3\). Then \(l=\sqrt{(0 - 0)^2+(-3 - 3)^2}=\sqrt{(-6)^2}=6\). For the width (shorter side) between (0,3) and (2,0), \(x_1 = 0,y_1 = 3,x_2 = 2,y_2 = 0\). Then \(w=\sqrt{(2 - 0)^2+(0 - 3)^2}=\sqrt{4 + 9}=\sqrt{13}\).

Step2: Calculate the perimeter

The perimeter formula of a rectangle is \(P = 2(l + w)\). Substituting \(l = 6\) and \(w=\sqrt{13}\), we get \(P=2(6+\sqrt{13})=12 + 2\sqrt{13}\) feet.

Step3: Calculate the area

The area formula of a rectangle is \(A=l\times w\). Substituting \(l = 6\) and \(w=\sqrt{13}\), we get \(A=6\sqrt{13}\) square - feet.

Answer:

(a) Length: 6 feet, Width: \(\sqrt{13}\) feet
(b) Perimeter: \(12 + 2\sqrt{13}\) feet, Area: \(6\sqrt{13}\) square - feet