# A triangle as corners at #(5 ,3 )#, #(1 ,4 )#, and #(3 ,5 )#. If the triangle is dilated by a factor of #3 # about #(2 ,2 ), how far will its centroid move?

The distance is

The distance between the 2 centroids is

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To find how far the centroid of the triangle moves when it is dilated by a factor of 3 about the point (2, 2), we need to first calculate the centroid of the original triangle and then find its new coordinates after dilation.

The centroid of a triangle is the point where the three medians intersect. The medians of a triangle are the lines joining each vertex to the midpoint of the opposite side.

For the original triangle with vertices (5, 3), (1, 4), and (3, 5), the midpoints of the sides are:

Midpoint of (5, 3) and (1, 4): [ \left( \frac{5 + 1}{2}, \frac{3 + 4}{2} \right) = \left( 3, \frac{7}{2} \right) ]

Midpoint of (1, 4) and (3, 5): [ \left( \frac{1 + 3}{2}, \frac{4 + 5}{2} \right) = \left( 2, \frac{9}{2} \right) ]

Midpoint of (3, 5) and (5, 3): [ \left( \frac{3 + 5}{2}, \frac{5 + 3}{2} \right) = \left( 4, 4 \right) ]

Now, we find the centroid, which is the average of the vertices:

Centroid: [ \left( \frac{5 + 1 + 3}{3}, \frac{3 + 4 + 5}{3} \right) = \left( \frac{9}{3}, \frac{12}{3} \right) = \left( 3, 4 \right) ]

After dilation by a factor of 3 about the point (2, 2), the new coordinates of the centroid will be three times the distance from the center of dilation to the original centroid, added to the coordinates of the center of dilation:

New centroid coordinates: [ x_{new} = 3 \times (2 - 3) + 2 = -1 + 2 = 1 ] [ y_{new} = 3 \times (2 - 4) + 2 = -6 + 2 = -4 ]

So, the centroid moves to the point (1, -4). Therefore, the centroid moves a distance of 5 units horizontally and 8 units vertically.

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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.

- A line segment has endpoints at #(2 ,7 )# and #(5 ,4 )#. The line segment is dilated by a factor of #2 # around #(4 ,3 )#. What are the new endpoints and length of the line segment?
- A line segment has endpoints at #(3 ,7 )# and #(4 ,9)#. If the line segment is rotated about the origin by #(pi )/2 #, translated vertically by #2 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- Points A and B are at #(9 ,7 )# and #(4 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?
- Circle A has a radius of #3 # and a center of #(5 ,9 )#. Circle B has a radius of #4 # and a center of #(1 ,2 )#. If circle B is translated by #<3 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(3 ,5 )# and #(2 ,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?

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