# A triangle has corners at #(7 ,3 )#, #(9 ,4 )#, and #(5 ,2 )#. If the triangle is dilated by a factor of #2 # about point #(6 ,1 ), how far will its centroid move?

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To find how far the centroid of the triangle moves when dilated by a factor of 2 about point (6, 1), follow these steps:

- Find the coordinates of the centroid of the original triangle.
- Dilate the original triangle by a factor of 2 about point (6, 1) to find the new coordinates of the vertices.
- Find the coordinates of the centroid of the dilated triangle.
- Calculate the distance between the centroids of the original and dilated triangles.

First, find the coordinates of the centroid of the original triangle:

Centroid coordinates = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)

Using the given coordinates: Centroid coordinates = ((7 + 9 + 5) / 3, (3 + 4 + 2) / 3) = (21 / 3, 9 / 3) = (7, 3)

Now, dilate the original triangle by a factor of 2 about point (6, 1):

For each vertex (x, y), the new coordinates are: New_x = 2 * (x - 6) + 6 New_y = 2 * (y - 1) + 1

Using the given vertices: Vertex 1: (7, 3) New_x1 = 2 * (7 - 6) + 6 = 8 New_y1 = 2 * (3 - 1) + 1 = 5

Vertex 2: (9, 4) New_x2 = 2 * (9 - 6) + 6 = 12 New_y2 = 2 * (4 - 1) + 1 = 7

Vertex 3: (5, 2) New_x3 = 2 * (5 - 6) + 6 = 4 New_y3 = 2 * (2 - 1) + 1 = 3

Now, find the coordinates of the centroid of the dilated triangle:

Centroid coordinates = ((New_x1 + New_x2 + New_x3) / 3, (New_y1 + New_y2 + New_y3) / 3)

Centroid coordinates = ((8 + 12 + 4) / 3, (5 + 7 + 3) / 3) = (24 / 3, 15 / 3) = (8, 5)

Finally, calculate the distance between the centroids of the original and dilated triangles:

Distance = √((x2 - x1)^2 + (y2 - y1)^2) = √((8 - 7)^2 + (5 - 3)^2) = √(1^2 + 2^2) = √(1 + 4) = √5

So, the centroid of the original triangle moves √5 units when the triangle is dilated by a factor of 2 about point (6, 1).

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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 with endpoints at #(5 , -2 )# and #(2, -7 )# is rotated clockwise by #pi/2#. What are the new endpoints of the line segment?
- Point A is at #(-8 ,2 )# and point B is at #(7 ,-1 )#. Point A is rotated #pi/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?
- Circle A has a radius of #3 # and a center of #(3 ,2 )#. Circle B has a radius of #5 # and a center of #(4 ,7 )#. If circle B is translated by #<2 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment goes from #(1 ,2 )# to #(4 ,1 )#. The line segment is reflected across #x=-1#, reflected across #y=3#, and then dilated about #(2 ,2 )# by a factor of #3#. How far are the new endpoints from the origin?
- A triangle has corners at #(2, 5 )#, #( 1, 3 )#, and #( 4 , 2 )#. If the triangle is dilated by # 7 x# around #(1, 3)#, what will the new coordinates of its corners be?

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