Question

consider polygon abcde on the coordinate grid what is the best approximation for the perimeter of polygon ABCDE

Answer 1

24.28 units

Step-by-step explanation

Perimeter is the distance around the edge of a shape, to find the perimeter of the given figure we can use the distance bewteen 2 points formula,it says

`\begin{gathered} d=√((x_2-x_1)^2+(y_2-y_1)^2) \\ where \\ P1(x_1,y_1) \\ P2(x_2,y_2) \end{gathered}`

so

Step 1

identify the vertices ;

`\begin{gathered} A(-2,3) \\ B(0,6) \\ C(5,4) \\ D(4,-1) \\ E(-1,-2) \end{gathered}`

Step 2

now, find the length of each segment

a)AB

replace in the formula

`\begin{gathered} d=√((x_2-x_1)^2+(y_2-y_1)^2) \\ d_(AB)=√((0-(-2))^2+(6-3)^2) \\ d_(AB)=√(4+9) \\ d_(AB)=√(13) \end{gathered}`

b)BC

`\begin{gathered} d=√((x_2-x_1)^2+(y_2-y_1)^2) \\ d_(BC)=√((5-0)^2+(4-6)^2) \\ d_(BC)=√(25+4) \\ d_(BC)=√(29) \end{gathered}`

c)CD

`\begin{gathered} d=√((x_2-x_1)^2+(y_2-y_1)^2) \\ d_(CD)=√((4-5)^2+(-1-4)^2) \\ d_(CD)=√(1+25) \\ d_(CD)=√(26) \end{gathered}`

d)DE

`\begin{gathered} d=√((x_2-x_1)^2+(y_2-y_1)^2) \\ d_(DE)=√((-1-4)^2+(-2-(-1))^2) \\ d_(DE)=√(25+1) \\ d_(DE)=√(26) \end{gathered}`

e) EA

`\begin{gathered} d=√((x_2-x_1)^2+(y_2-y_1)^2) \\ d_(EA)=√((-2-(-1))^2+(3-(-2))^2) \\ d_(EA)=√(1+25) \\ d_(EA)=√(26) \end{gathered}`

Step 3

finally, the perimeter is the sum of the sides length, so

`\begin{gathered} perimeter=AB+BC+CD+DE+EA \\ replacing \\ perimeter=√(13)+√(29)+√(26)+√(26)+√(26) \\ perimeter=24.28 \end{gathered}`

so, the perimeter is 24.28 units