What are the coordinates of the centroid of a triangle with vertices X(−4, 0) , Y(−1, 4) , and Z(2, 2) ? Enter your answer in the boxes.

[tex]distance \: xy \: = \sqrt{( { (3)}^{2} + {(4)}^{2} ) } \\ = \sqrt{(9 + 16)} = \sqrt{25} = 5[/tex]
[tex]distance \: xz \: = \sqrt{( {(2)}^{2} + {(6)}^{2} ) } \\ = \sqrt{(4 + 36)} = \sqrt{40} \\ = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} [/tex]

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MrRoyal MrRoyal · Answer 2 · 2021-10-15T14:17:59+02:00

The centroid of a triangle is the point where the medians of the triangle meets.

The coordinates of the centroid is (-1,2)

The coordinates of the triangle are given as:

[tex]X = (-4,0)[/tex]

[tex]Y = (-1,4)[/tex]

[tex]Z = (2,2)[/tex]

The coordinates of the centroid is:

[tex]C = \frac 13(x_1 + x_2 + x_3,y_1+y_2+y_3)[/tex]

So, we have:

[tex]C = \frac 13(-4-1 + 2,0+4+2)[/tex]

[tex]C = \frac 13(-3,6)[/tex]

Expand

[tex]C = (\frac 13\times -3,\frac 13\times 6)[/tex]

[tex]C = (-1,2)[/tex]

Hence, the coordinates of the centroid is (-1,2)