Respuesta :
Answer:
Explanation:
Since there are six points, the minimum distance from all points would be the centroid of polygon formed by A,B,C,D,E,F
To find the coordinates of centroid of a polygon we use the following formula. Let A be area of the polygon.
[tex]C_{x}=\frac{1}{6A} sum(({x_{i} +x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}))[/tex] where i=1 to N-1 and N=6
[tex]C_{y}=\frac{1}{6A} sum(({y_{i} +y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}))[/tex]
A area of the polygon can be found by the following formula[tex]A=\frac{1}{2} sum(x_{i} y_{i+1} -x_{i+1} y_{i})[/tex] where i=1 to N-1
[tex]A=\frac{1}{2}[ (x_{1} y_{2} -x_{2} y_{1})+ (x_{2} y_{3} -x_{3} y_{2})+(x_{3} y_{4} -x_{4} y_{3})+(x_{4} y_{5} -x_{5} y_{4})+(x_{5} y_{6} -x_{6} y_{5})][/tex]
A=0.5[(20×25 -25×15) +(25×32 -13×25)+(13×21 -4×32)+(4×8 -18×21)+(18×14 -25×8)
A=225.5 miles²
Now putting the value of area in Cx and Cy
[tex]C_{x} =\frac{1}{6A}[ [(x_{1}+x_{2})(x_{1} y_{2} -x_{2} y_{1})]+ [(x_{2}+x_{3})(x_{2} y_{3} -x_{3} y_{2})]+[(x_{3}+x_{4})(x_{3} y_{4} -x_{4} y_{3})]+[(x_{4}+x_{5})(x_{4} y_{5} -x_{5} y_{4})]+[(x_{5}+x_{6})(x_{5} y_{6} -x_{6} y_{5})]][/tex]
putting the values of x's and y's you will get
[tex]C_{x} =15.36[/tex]
For Cy
[tex]C_{y} =\frac{1}{6A}[ [(y_{1}+y_{2})(x_{1} y_{2} -x_{2} y_{1})]+ [(y_{2}+y_{3})(x_{2} y_{3} -x_{3} y_{2})]+[(y_{3}+y_{4})(x_{3} y_{4} -x_{4} y_{3})]+[(y_{4}+y_{5})(x_{4} y_{5} -x_{5} y_{4})]+[(y_{5}+y_{6})(x_{5} y_{6} -x_{6} y_{5})]][/tex]
putting the values of x's and y's you will get
[tex]C_{y} =22.55[/tex]
So coordinates for the fire station should be (15.36,22.55)
In this exercise we have to use the knowledge of coordinates to calculate the points of coordinates X and Y, in this way we find that:
So coordinates will be [tex]15.36,22.55[/tex]
Since skilled happen a goal in football, the minimum distance from all points hopeful the centroid of closed plane figure make by A,B,C,D,E,F. To find the match of centroid of a closed plane figure we use the following recipe. Let A exist scope of a surface of the closed plane figure.
[tex]C_{x}=\frac{1}{6 A} \sum}\left(\left(x_{i}+x_{i+1}\right)\left(x_{i} y_{i+1}-x_{i+1} y_{i}\right)\right)[/tex] where i=1 to N-1 and N=6
[tex]C_{y}=\frac{1}{6 A} \sum}\left(\left(y_{i}+y_{i+1}\right)\left(x_{i} y_{i+1}-x_{i+1} y_{i}\right)\right)[/tex]
A area of the polygon can be found by the following formula:
[tex]A=\frac{1}{2} \sum}\left(x_{i} y_{i+1}-x_{i+1} y_{i}\right) \ where\ i=1 \ to \ N-1[/tex]
[tex]A=\frac{1}{2}\left[\left(x_{1} y_{2}-x_{2} y_{1}\right)+\left(x_{2} y_{3}-x_{3} y_{2}\right)+\left(x_{3} y_{4}-x_{4} y_{3}\right)+\left(x_{4} y_{5}-x_{5} y_{4}\right)+\left(x_{5} y_{6}-x_{6} y_{5}\right)\right][/tex]
[tex]A=0.5[(20*25 -25*15) +(25*32 -13*25)+(13*21 -4*32)+(4*8 -18*21)+(18*14 -25*8)[/tex]
[tex]A=225.5 \ miles[/tex]
Now putting the value of area in Cx, we have:
[tex]$C_{x}=\frac{1}{6 A}\left[\left[\left(x_{1}+x_{2}\right)\left(x_{1} y_{2}-x_{2} y_{1}\right)\right]+\left[\left(x_{2}+x_{3}\right)\left(x_{2} y_{3}-x_{3} y_{2}\right)\right]+\left[\left(x_{3}+x_{4}\right)\left(x_{3} y_{4}-\right.\right.\right.$\left.\left.\left.x_{4} y_{3}\right)\right]+\left[\left(x_{4}+x_{5}\right)\left(x_{4} y_{5}-x_{5} y_{4}\right)\right]+\left[\left(x_{5}+x_{6}\right)\left(x_{5} y_{6}-x_{6} y_{5}\right)\right]\right]$[/tex]
Putting the values of x's and y's you will get
[tex]$C_{x}=15.36$[/tex]
For Cy, we have that:
[tex]$C_{y}=\frac{1}{6 A}\left[\left[\left(y_{1}+y_{2}\right)\left(x_{1} y_{2}-x_{2} y_{1}\right)\right]+\left[\left(y_{2}+y_{3}\right)\left(x_{2} y_{3}-x_{3} y_{2}\right)\right]+\left[\left(y_{3}+y_{4}\right)\left(x_{3} y_{4}-\right.\right.\right.$\left.\left.\left.x_{4} y_{3}\right)\right]+\left[\left(y_{4}+y_{5}\right)\left(x_{4} y_{5}-x_{5} y_{4}\right)\right]+\left[\left(y_{5}+y_{6}\right)\left(x_{5} y_{6}-x_{6} y_{5}\right)\right]\right]$[/tex]
Putting the values of x's and y's you will get
[tex]C_y= 22.55[/tex]
See more about coordinates at brainly.com/question/17514638