Question 149703

The formula for the coordinates of the centroid is 


{{{x=(1/3)(x[a]+x[b]+x[c])}}} and {{{y=(1/3)(y[a]+y[b]+y[c])}}}


note: notice how we're simply averaging the coordinates


where *[Tex \LARGE \left(x_{a},y_{a}\right) ] ,  *[Tex \LARGE \left(x_{b},y_{b}\right) ], and  *[Tex \LARGE \left(x_{c},y_{c}\right) ]  are the coordinates of the three vertices a, b, and c respectively.



a)


{{{x=(1/3)(x[a]+x[b]+x[c])}}} Start with the formula for finding the x-coordinate of the centroid.



{{{x=(1/3)(-1-2+3)}}} Plug in the x-coordinates of the given points



{{{x=(1/3)(0)}}} Add



{{{x=0}}} Multiply.



So the x-coordinate of the centroid is {{{x=0}}}


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{{{y=(1/3)(y[a]+y[b]+y[c])}}} Start with the formula for finding the y-coordinate of the centroid.



{{{y=(1/3)(5+8+3)}}} Plug in the x-coordinates of the given points



{{{y=(1/3)(16)}}} Add



{{{y=5.33}}} Simplify.



So the y-coordinate of the centroid is {{{y=5.33}}}




So the centroid is (0,5.33)


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b)




{{{x=(1/3)(x[a]+x[b]+x[c])}}} Start with the formula for finding the x-coordinate of the centroid.



{{{x=(1/3)(2+8+14)}}} Plug in the x-coordinates of the given points



{{{x=(1/3)(24)}}} Add



{{{x=8}}} Multiply.



So the x-coordinate of the centroid is {{{x=8}}}


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(2,7),(8,1) and (14,11)

{{{y=(1/3)(y[a]+y[b]+y[c])}}} Start with the formula for finding the y-coordinate of the centroid.



{{{y=(1/3)(7+1+11)}}} Plug in the x-coordinates of the given points



{{{y=(1/3)(19)}}} Add



{{{y=6.33}}} Simplify.



So the y-coordinate of the centroid is {{{y=6.33}}}



So the centroid is (0,5.33)




<hr>



c)


(a,p), (b,q) and (c,r).

{{{x=(1/3)(x[a]+x[b]+x[c])}}} Start with the formula for finding the x-coordinate of the centroid.



{{{x=(1/3)(a+b+c)}}} Plug in the x-coordinates of the given points




So the x-coordinate of the centroid is {{{x=(1/3)(a+b+c)}}}


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{{{x=(1/3)(x[a]+x[b]+x[c])}}} Start with the formula for finding the x-coordinate of the centroid.



{{{x=(1/3)(p+q+r)}}} Plug in the x-coordinates of the given points



So the y-coordinate of the centroid is {{{y=(1/3)(p+q+r)}}}



So the centroid is ({{{(1/3)(a+b+c)}}},{{{(1/3)(p+q+r)}}})