# A triangle has corners at #(5 ,3 )#, #(4 ,6 )#, and #(8 ,5 )#. If the triangle is dilated by a factor of #2 # about point #(3 ,2 )#, how far will its centroid move?

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The centroid of a triangle moves according to the scale factor during dilation. To find how far the centroid moves, we first need to find the centroid of the original triangle and then determine its new position after dilation.

The centroid of a triangle is calculated by finding the average of the coordinates of its vertices.

Original triangle: (A(5, 3)), (B(4, 6)), (C(8, 5))

New centroid after dilation: (G'\left(\frac{1}{3}(5+4+8), \frac{1}{3}(3+6+5)\right))

Then, we find the distance between the original centroid and the new centroid.

The original centroid is ((G_x, G_y) = \left(\frac{1}{3}(5+4+8), \frac{1}{3}(3+6+5)\right)).

The new centroid is (G'\left(\frac{1}{3}(2\cdot5+3), \frac{1}{3}(2\cdot3+2)\right)).

Find the distance between these two points using the distance formula:

(d = \sqrt{(G'_x - G_x)^2 + (G'_y - G_y)^2})

(d = \sqrt{\left(\frac{1}{3}(2\cdot5+3) - \frac{1}{3}(5+4+8)\right)^2 + \left(\frac{1}{3}(2\cdot3+2) - \frac{1}{3}(3+6+5)\right)^2})

(d = \sqrt{\left(\frac{1}{3}(13) - \frac{1}{3}(17)\right)^2 + \left(\frac{1}{3}(8) - \frac{1}{3}(14)\right)^2})

(d = \sqrt{\left(\frac{13}{3} - \frac{17}{3}\right)^2 + \left(\frac{8}{3} - \frac{14}{3}\right)^2})

(d = \sqrt{\left(\frac{-4}{3}\right)^2 + \left(\frac{-6}{3}\right)^2})

(d = \sqrt{\frac{16}{9} + \frac{36}{9}})

(d = \sqrt{\frac{52}{9}})

So, the distance the centroid moves after dilation is (\frac{\sqrt{52}}{3}).

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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.

- Points A and B are at #(7 ,1 )# and #(7 ,5 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # and dilated about point C by a factor of #1/2 #. If point A is now at point B, what are the coordinates of point C?
- A line segment has endpoints at #(3 ,7 )# and #(4 ,5)#. If the line segment is rotated about the origin by #(pi )/2 #, translated vertically by #-1 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- Points A and B are at #(4 ,3 )# and #(1 ,4 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?
- A line segment has endpoints at #(3 ,8 )# and #(4 ,6)#. If the line segment is rotated about the origin by #(pi )/2 #, translated vertically by #3 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- A line segment has endpoints at #(4 ,9 )# and #(5 ,2)#. If the line segment is rotated about the origin by #pi #, translated vertically by #-4 #, and reflected about the x-axis, what will the line segment's new endpoints be?

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