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

Distance moved by centroid is

Coordinates of centroid is obtained using the formula,

Triangle is dilated about point D(-1,3) by a factor of 2/5

Using distance formula, we can calculate the distance moved by centroid

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First, find the coordinates of the centroid of the original triangle using the formula: [ \text{Centroid} = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]

Original triangle's centroid: ((\frac{2 - 1 + 3}{3}, \frac{-5 + 2 - 3}{3}) = (\frac{4}{3}, -2))

Next, find the coordinates of the centroid of the dilated triangle using the formula: [ \text{New Centroid} = \left( \frac{2}{5} \cdot (x_{\text{centroid}} - x_{\text{dilation}}) + x_{\text{dilation}}, \frac{2}{5} \cdot (y_{\text{centroid}} - y_{\text{dilation}}) + y_{\text{dilation}} \right) ]

Substitute the values: [ \text{New Centroid} = \left( \frac{2}{5} \cdot (\frac{4}{3} - (-1)) + (-1), \frac{2}{5} \cdot (-2 - 3) + 3 \right) ] [ \text{New Centroid} = \left( \frac{2}{5} \cdot \frac{7}{3} - \frac{5}{5}, \frac{2}{5} \cdot (-5) + 3 \right) ] [ \text{New Centroid} = \left( \frac{14}{15} - \frac{5}{5}, -\frac{10}{5} + 3 \right) ] [ \text{New Centroid} = \left( \frac{14}{15} - 1, -2 + 3 \right) ] [ \text{New Centroid} = \left( \frac{-1}{15}, 1 \right) ]

So, the centroid of the dilated triangle moves (\left( \frac{-1}{15}, 1 \right)) units from its original position.

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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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