Question

example: a new community garden has four corners. starting at the first corner and working counterclockwise, the second corner is 200 feet east, the third corner is 150 feet north of the second corner, and the fourth corner is 100 feet west of the third corner. represent the garden in the coordinate plane and determine how much fence is needed for the perimeter of the garden, as well as the total area of the garden. ***round to the nearest tenth when needed

Explanation:

Step1: Place points on coordinate - plane

Let the first corner be at the origin $(0,0)$. The second corner is at $(200,0)$ (200 feet east). The third corner is at $(200,150)$ (150 feet north of the second corner). The fourth corner is at $(100,150)$ (100 feet west of the third corner).

Step2: Calculate the perimeter

The lengths of the sides of the quadrilateral:

• The first - side length from $(0,0)$ to $(200,0)$ is $d_1=\sqrt{(200 - 0)^2+(0 - 0)^2}=200$ feet.
• The second - side length from $(200,0)$ to $(200,150)$ is $d_2=\sqrt{(200 - 200)^2+(150 - 0)^2}=150$ feet.
• The third - side length from $(200,150)$ to $(100,150)$ is $d_3=\sqrt{(100 - 200)^2+(150 - 150)^2}=100$ feet.
• The fourth - side length from $(100,150)$ to $(0,0)$ is $d_4=\sqrt{(0 - 100)^2+(0 - 150)^2}=\sqrt{10000 + 22500}=\sqrt{32500}=50\sqrt{13}\approx180.3$ feet.

The perimeter $P=d_1 + d_2 + d_3 + d_4=200+150 + 100+180.3=630.3$ feet.

Step3: Calculate the area

The garden is a trapezoid. The formula for the area of a trapezoid is $A=\frac{(b_1 + b_2)h}{2}$, where $b_1$ and $b_2$ are the lengths of the parallel sides and $h$ is the height between them. Here, $b_1 = 200$ feet, $b_2 = 100$ feet, and $h = 150$ feet.
$A=\frac{(200 + 100)\times150}{2}=\frac{300\times150}{2}=22500$ square feet.

Answer:

Perimeter: $630.3$ feet; Area: $22500$ square feet