Question

consider polygon abcde on the coordinate grid.
what is the best approximation for the perimeter of polygon abcde?
a. 24.3 units
b. 11.0 units
c. 17.0 units
d. 22.1 units

Explanation:

Step1: Find the coordinates of each point

Assume \(A(-1,3)\), \(B(0,6)\), \(C(5,3)\), \(D(4,-1)\), \(E(-1,-1)\)

Step2: Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

• For \(AB\): \(x_1=-1,y_1 = 3,x_2=0,y_2 = 6\)

\(d_{AB}=\sqrt{(0 + 1)^2+(6 - 3)^2}=\sqrt{1 + 9}=\sqrt{10}\approx3.16\)

• For \(BC\): \(x_1=0,y_1 = 6,x_2=5,y_2 = 3\)

\(d_{BC}=\sqrt{(5 - 0)^2+(3 - 6)^2}=\sqrt{25 + 9}=\sqrt{34}\approx5.83\)

• For \(CD\): \(x_1=5,y_1 = 3,x_2=4,y_2 = -1\)

\(d_{CD}=\sqrt{(4 - 5)^2+(-1 - 3)^2}=\sqrt{1+16}=\sqrt{17}\approx4.12\)

• For \(DE\): \(x_1=4,y_1 = -1,x_2=-1,y_2 = -1\)

\(d_{DE}=\sqrt{(-1 - 4)^2+(-1+1)^2}=\sqrt{25+0}=5\)

• For \(EA\): \(x_1=-1,y_1 = -1,x_2=-1,y_2 = 3\)

\(d_{EA}=\sqrt{(-1 + 1)^2+(3 + 1)^2}=\sqrt{0 + 16}=4\)

Step3: Calculate the perimeter \(P\)

\(P=d_{AB}+d_{BC}+d_{CD}+d_{DE}+d_{EA}\)
\(P\approx3.16+5.83+4.12+5+4\)
\(P\approx22.11\)

Answer:

D. 22.1 units