A triangle has corners at #(5 ,2 )#, #(-3 ,-3 )#, and #(1 ,-2 )#. If the triangle is dilated by a factor of #1/3 # about point #(5 ,-2 ), how far will its centroid move?
The centroid will move by
The distance between the centroids is
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The centroid of a triangle is the point of intersection of its medians. When a triangle is dilated by a factor of ( k ) about a point, the centroid moves in the same direction as the original centroid, but the distance is scaled by ( k ).
To find the centroid of the original triangle, we calculate the average of the coordinates of its vertices.
Centroid of the original triangle: [ x_{\text{centroid}} = \frac{5 - 3 + 1}{3} = \frac{3}{3} = 1 ] [ y_{\text{centroid}} = \frac{2 - 3 - 2}{3} = \frac{-3}{3} = -1 ]
The centroid of the original triangle is at (1, -1).
When dilated by a factor of ( \frac{1}{3} ) about point (5, -2), the centroid will move ( \frac{1}{3} ) of the distance from the point of dilation to the original centroid.
Distance from (1, -1) to (5, -2): [ \sqrt{(5 - 1)^2 + (-2 - (-1))^2} = \sqrt{16 + 1} = \sqrt{17} ]
So, the distance the centroid moves is: [ \frac{1}{3} \times \sqrt{17} ]
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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.
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 triangle has corners at #(2, 7 )#, #( 8, 3 )#, and #( 4 , 8 )#. If the triangle is dilated by # 7 x# around #(1, 3)#, what will the new coordinates of its corners be?
- Circle A has a radius of #2 # and a center at #(8 ,3 )#. Circle B has a radius of #3 # and a center at #(3 ,5 )#. If circle B is translated by #<-2 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(0 ,0 )# and #(1 , 3 )#. If the line segment is rotated about the origin by # pi /2 #, translated horizontally by # 5 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #1 # and a center of #(5 ,2 )#. Circle B has a radius of #2 # and a center of #(4 ,5 )#. If circle B is translated by #<-3 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(5 ,2 )# and #(3 ,4 )#. If the line segment is rotated about the origin by #(3 pi)/2 #, translated vertically by #4 #, and reflected about the y-axis, what will the line segment's new endpoints be?

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