Question

modeling real life the diagram shows the vertices of a lion sanctuary. each unit in the coordinate plane represents 100 feet. find the perimeter and the area of the sanctuary. a(5, 7) b(8, 7) e(1, 4) f(5, 4) d(1, 1) c(8, 1) perimeter: ft area: ft²

Explanation:

Step1: Calculate side - lengths

Use the distance formula for horizontal and vertical lines. For horizontal lines \(y = k\), the distance between \((x_1,k)\) and \((x_2,k)\) is \(d=\vert x_2 - x_1\vert\), and for vertical lines \(x = h\), the distance between \((h,y_1)\) and \((h,y_2)\) is \(d=\vert y_2 - y_1\vert\).
\(DE=\vert4 - 1\vert= 3\) units, \(EF=\vert5 - 1\vert = 4\) units, \(FA=\vert7 - 4\vert=3\) units, \(AB=\vert8 - 5\vert = 3\) units, \(BC=\vert7 - 1\vert=6\) units, \(CD=\vert8 - 1\vert = 7\) units.
Since each unit represents 100 feet, the actual side - lengths are: \(DE = 3\times100=300\) feet, \(EF = 4\times100 = 400\) feet, \(FA=3\times100 = 300\) feet, \(AB = 3\times100=300\) feet, \(BC=6\times100 = 600\) feet, \(CD=7\times100 = 700\) feet.

Step2: Calculate the perimeter

The perimeter \(P\) is the sum of all side - lengths.
\(P=300 + 400+300 + 300+600+700=2600\) feet.

Step3: Calculate the area

We can divide the figure into two rectangles. One rectangle with vertices \(D(1,1)\), \(E(1,4)\), \(F(5,4)\), \(C(5,1)\) and another with vertices \(F(5,4)\), \(A(5,7)\), \(B(8,7)\), \(C(8,4)\).
The area of the first rectangle with length \(l_1=5 - 1 = 4\) units and width \(w_1=4 - 1=3\) units is \(A_1=4\times3\times100\times100 = 120000\) square feet.
The area of the second rectangle with length \(l_2=8 - 5 = 3\) units and width \(w_2=7 - 4 = 3\) units is \(A_2=3\times3\times100\times100=90000\) square feet.
The total area \(A=A_1 + A_2=120000+90000 = 210000\) square feet.

Answer:

Perimeter: 2600 ft
Area: 210000 ft²