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Chris L T521
Gold Member
MHB
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Thanks to those who participated in last week's POTW! Here's this week's problem!
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Problem: Let $(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)$ be points in $\mathbb{R}^2$ such that when they are joined together in succession by line segments [and $(x_n,y_n)$ is joined to $(x_1,y_1)$], the segments enclose a polygonal region $R$ (see attached picture).
Assuming that the polygonal boundary is counterclockwise oriented, prove that the area of the region $R$ is given in terms of the coordinates of its vertices as follows:
\[A(R)=\frac{1}{2}\left(\begin{vmatrix}x_1 & x_2\\ y_1 & y_2\end{vmatrix}+\begin{vmatrix}x_2 & x_3\\ y_2 & y_3\end{vmatrix}+\cdots+\begin{vmatrix}x_{n-1} & x_n\\ y_{n-1} & y_n\end{vmatrix}+\begin{vmatrix}x_n & x_1\\ y_n & y_1\end{vmatrix}\right)\]
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Hint:
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Problem: Let $(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)$ be points in $\mathbb{R}^2$ such that when they are joined together in succession by line segments [and $(x_n,y_n)$ is joined to $(x_1,y_1)$], the segments enclose a polygonal region $R$ (see attached picture).
Assuming that the polygonal boundary is counterclockwise oriented, prove that the area of the region $R$ is given in terms of the coordinates of its vertices as follows:
\[A(R)=\frac{1}{2}\left(\begin{vmatrix}x_1 & x_2\\ y_1 & y_2\end{vmatrix}+\begin{vmatrix}x_2 & x_3\\ y_2 & y_3\end{vmatrix}+\cdots+\begin{vmatrix}x_{n-1} & x_n\\ y_{n-1} & y_n\end{vmatrix}+\begin{vmatrix}x_n & x_1\\ y_n & y_1\end{vmatrix}\right)\]
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Hint:
Use Green's Theorem.
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