Question

1. polygon

abcde
a(-3,2)
b(7,2)
c(7,-3)
d(2,-8)
e(-3,-3)

Explanation:

Step1: Calculate the length of \(AB\)

Since \(A(-3,2)\) and \(B(7,2)\) have the same \(y\) - coordinate.
Use the distance formula \(d=\vert x_2 - x_1\vert\).
\(AB=\vert7-(-3)\vert=\vert7 + 3\vert = 10\)

Step2: Calculate the length of \(BC\)

Since \(B(7,2)\) and \(C(7,-3)\) have the same \(x\) - coordinate.
Use the distance formula \(d=\vert y_2 - y_1\vert\).
\(BC=\vert-3 - 2\vert=\vert-5\vert = 5\)

Step3: Calculate the length of \(CD\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \(C(7,-3)\) and \(D(2,-8)\)
\(CD=\sqrt{(2 - 7)^2+(-8+3)^2}=\sqrt{(-5)^2+(-5)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}\)

Step4: Calculate the length of \(DE\)

Since \(D(2,-8)\) and \(E(-3,-3)\)
\(DE=\sqrt{(-3 - 2)^2+(-3 + 8)^2}=\sqrt{(-5)^2+5^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}\)

Step5: Calculate the length of \(EA\)

Since \(E(-3,-3)\) and \(A(-3,2)\) have the same \(x\) - coordinate.
\(EA=\vert2-(-3)\vert=\vert2 + 3\vert=5\)

Step6: Calculate the perimeter

Perimeter \(P=AB + BC+CD + DE+EA\)
\(P=10 + 5+5\sqrt{2}+5\sqrt{2}+5=20 + 10\sqrt{2}\approx20+14.14 = 34.14\)

Step7: Calculate the area

Divide the polygon into a rectangle \(ABCE\) and a triangle \(CDE\)
Area of rectangle \(ABCE\): \(AB\times BE\), \(AB = 10\), \(BE=\vert2-(-3)\vert = 5\), \(A_{ABCE}=10\times5 = 50\)
Area of triangle \(CDE\): \(A=\frac{1}{2}\times base\times height\), base \(CD = 5\sqrt{2}\), height \(h = 5\sqrt{2}\) (using the formula for the area of a right - angled triangle, since the legs of the right - angled triangle formed by the differences in coordinates are equal)
\(A_{CDE}=\frac{1}{2}\times5\sqrt{2}\times5\sqrt{2}=\frac{1}{2}\times50 = 25\)
Total area \(A=50 + 25=75\)

Answer:

Perimeter: \(20 + 10\sqrt{2}\approx34.14\); Area: \(75\)