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The vertices of a convex pentagon are (-1,-1), (-3,4), (1,7), (6,5), and (3,-1). What is the area of the pentagon 

 Feb 25, 2020
 #1
avatar
+1

you can use shoelace

 

from there you would get

 

-1 -1

-3  4

1   7

6   5

3  -1

1  -1

 

so the answer is 47

 Feb 25, 2020
 #3
avatar+25529 
+2

The vertices of a convex pentagon are (-1,-1), (-3,4), (1,7), (6,5), and (3,-1).
What is the area of the pentagon .

 

Use the Gauss's shoelace area formula. Fill the table in counterclockwise.

\(\begin{array}{|lll|} \hline \mathbf{\text{The area of a convex pentagon } }: \\ \begin{array}{r|rr|c} \hline \text{Point} & x & y & \text{cross products} \\ \hline (-1,~-1): & -1 & -1 \\ & & & (-1)*(-1) -(3)*(-1) \\ (3,~-1): & 3 & -1 \\ & & & 3*5 -6*(-1) \\ (6,~5): & 6 & 5 & \\ & & & 6*7-1*5 \\ (1,~7): & 1 & 7 & \\ & & & 1*4 -(-3)*7 \\ (-3,~4): & -3 & 4 & \\ & & & (-3)*(-1) -(-1)*4 \\ (-1,~-1): & -1 & -1 & \\ \hline & & & \text{sum }~=1+3+15+6+42-5+4+21+3+4 \\ & & & \mathbf{=94} \\ \hline \end{array}\\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline \text{The area of the convex pentagon} &=& \dfrac{1}{2}*\text{sum} \\\\ &=& \dfrac{94}{2} \\\\ &=& \mathbf{47} \\ \hline \end{array}\)

 

Source "Gauss's magic shoelace area formula and its calculus companion" see:
https://www.youtube.com/watch?v=0KjG8Pg6LGk

 

laugh

 Feb 25, 2020
 #4
avatar+110227 
+1

Hi Heureka,

Thanks for your great answer.   

 

Your answer was initially flagged for moderation because of your youtube link.

I am just saying this so you will understand what the problem is next time.   indecision  laugh

 

Thanks to guest too although your answer is cryptic.

Melody  Feb 25, 2020
edited by Melody  Feb 25, 2020
 #5
avatar+25529 
+2

Hi Melody,

 

many thanks for the detailed answer!

 

laugh

heureka  Feb 25, 2020

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