# A triangle has corners at #(2 ,-2 )#, #(3 ,-1 )#, and #(5 ,7 )#. If the triangle is dilated by a factor of #2/5 # about point #(6 ,-2 ), how far will its centroid move?

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The centroid of a triangle is a point where the three medians of the triangle intersect. In a dilation, the centroid moves along with the dilation center. Given that the triangle is dilated by a factor of 2/5 about point (6, -2), the centroid will move proportionally to the dilation.

To find the centroid of the original triangle, you calculate the average of the coordinates of its vertices. Then, apply the dilation factor to determine how far the centroid moves.

Let's denote the original triangle's vertices as A(2, -2), B(3, -1), and C(5, 7).

The coordinates of the centroid (G) of the original triangle can be calculated as:

[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) ]

Plugging in the coordinates:

[ G = \left(\frac{2 + 3 + 5}{3}, \frac{-2 - 1 + 7}{3}\right) ]

[ G = \left(\frac{10}{3}, \frac{4}{3}\right) ]

Now, to find how far the centroid moves after the dilation, you use the dilation factor. The new coordinates of the centroid (G') can be calculated as:

[ G' = \left(x_c + \frac{2}{5}(x_g - x_c), y_c + \frac{2}{5}(y_g - y_c)\right) ]

Where: ( (x_c, y_c) ) = coordinates of the dilation center (6, -2) ( (x_g, y_g) ) = coordinates of the original centroid (10/3, 4/3)

Plugging in the values:

[ G' = \left(6 + \frac{2}{5}\left(\frac{10}{3} - 6\right), -2 + \frac{2}{5}\left(\frac{4}{3} + 2\right)\right) ]

[ G' = \left(6 + \frac{2}{5}\left(\frac{10}{3} - 6\right), -2 + \frac{2}{5}\left(\frac{4}{3} + 6\right)\right) ]

[ G' = \left(6 + \frac{2}{5}\left(\frac{10 - 18}{3}\right), -2 + \frac{2}{5}\left(\frac{4 + 18}{3}\right)\right) ]

[ G' = \left(6 + \frac{2}{5}\left(-\frac{8}{3}\right), -2 + \frac{2}{5}\left(\frac{22}{3}\right)\right) ]

[ G' = \left(6 - \frac{16}{15}, -2 + \frac{44}{15}\right) ]

[ G' = \left(\frac{90 - 16}{15}, \frac{-30 + 44}{15}\right) ]

[ G' = \left(\frac{74}{15}, \frac{14}{15}\right) ]

So, the centroid of the dilated triangle moves to the point approximately (4.933, 0.933).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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