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FromTheMoon [43]
3 years ago
6

Find the perimeter of the polygon with vertices A(-6,-4), B(-3,6), C(4,0), and D(2,-1)?

Mathematics
1 answer:
EleoNora [17]3 years ago
6 0
Check the picture below.

so the perimeter of the polygon is the sum of all its sides, namely, AB + BC + CD + DA.

now, let's check how long each side is,

\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\
\begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&A&(~{{ -6}} &,&{{ -4}}~) 
%  (c,d)
&B&(~{{ -3}} &,&{{ 6}}~)
\end{array}
\\\\\\
d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2}\\\\
-------------------------------\\\\
AB=\sqrt{[-3-(-6)]^2+[6-(-4)]^2}
\\\\\\
AB=\sqrt{(-3+6)^2+(6+4)^2}
\\\\\\
AB=\sqrt{3^2+10^2}\implies \boxed{AB=\sqrt{109}}\\\\
-------------------------------

\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\
\begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&B&(~{{ -3}} &,&{{6}}~) 
%  (c,d)
&C&(~{{ 4}} &,&{{ 0}}~)
\end{array}
\\\\
-------------------------------\\\\
BC=\sqrt{[4-(-3)]^2+[0-6]^2}\implies BC=\sqrt{(4+3)^2+(0-6)^2}
\\\\\\
BC=\sqrt{7^2+(-6)^2}\implies \boxed{BC=\sqrt{85}}\\\\
-------------------------------

\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\
\begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&C&(~{{ 4}} &,&{{0}}~) 
%  (c,d)
&D&(~{{ 2}} &,&{{ -1}}~)
\end{array}
\\\\
-------------------------------\\\\
CD=\sqrt{(2-4)^2+(-1-0)^2}\implies CD=\sqrt{(-2)^2+(-1)^2}
\\\\\\
\boxed{CD=\sqrt{5}}\\\\
-------------------------------

\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\
\begin{array}{ccccccccc}
&&x_1&&y_1&&x_2&&y_2\\
%  (a,b)
&D(~{{ 2}} &,&{{-1}}~) 
%  (c,d)
&A&(~{{ -6}} &,&{{ -4}}~)
\end{array}\\\\
-------------------------------\\\\
DA=\sqrt{[-6-2]^2+[-4-(-1)]^2}\\\\\\ DA=\sqrt{(-6-2)^2+(-4+1)^2}
\\\\\\
DA=\sqrt{(-8)^2+(-3)^2}\implies \boxed{DA=\sqrt{73}}

sum those sides up, and that's the perimeter of the polygon.

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