Using the shoelace formula (Green’s Theorem) in counterclockwise fashion, the value of x is -8.
Area of Polygon can be calculated using Shoelace formula described by Mathematician and Physicist Carl Friedrich Gauss where polygon vertices are described by their Cartesian coordinates in the Cartesian plane.
A = [tex]\frac{1}{2} |x_{1} y_{2} +x_{2} y_{3}+........x_{n-1}y_{n} +x_{n} y_{1} -x_{2} y_{1} -x_{3} y_{2} - ........-x_{n}y_{n-1}-x_{2}y_{n} |[/tex]
Shoelace formula (Green’s Theorem)
A = [tex]\frac{1}{2} |x_{1} y_{2} +x_{2} y_{3}+........x_{n-1}y_{n} +x_{n} y_{1} -x_{2} y_{1} -x_{3} y_{2} - ........-x_{n}y_{n-1}-x_{2}y_{n} |[/tex]
Where A is the area of the polygon and n is the number of sides of the polygon.
Here A = 6, n = 3, [tex]x_{1}[/tex] = 6, [tex]y_{1}[/tex] = 0, [tex]x_{2}[/tex] = 5, [tex]y_{2}[/tex] = 10, [tex]x_{3}[/tex] = [tex]x[/tex], [tex]y_{3}[/tex] = 20
A = 60 = [tex]\frac{1}{2} |(6.10)+(5.20)+(x.0)-(5.0)-(x.0)-(6.20)|[/tex]
60 = [tex]\frac{1}{2} |60+100-10x-120|[/tex]
60 = [tex]\frac{1}{2} |40-10x|[/tex]
60 = [tex]|20-5x|[/tex]
So 20 - 5x = -60
5x = 80
x = 16
20 - 5x = 60
5x = -40
x = -8
since x is a negative value,
we can know x = -8
Hence the value of x is -8.
Find out more information about green's theorem here
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