A triangle has corners at #(3 ,1 )#, #(1 ,-2 )#, and #(-4 ,-5 )#. If the triangle is dilated by a factor of #5 # about point #(7 ,-6 ), how far will its centroid move?
The centroid is the center of mass of the triangle and is given by the average of the coordinates:
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To find the centroid of the triangle before and after dilation, follow these steps:
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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) ]
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Dilate the coordinates of the centroid using the given dilation factor and center.
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Calculate the displacement of the centroid by finding the difference between the dilated centroid and the original centroid.
Let's calculate:
Original triangle coordinates: ( A(3, 1) ) ( B(1, -2) ) ( C(-4, -5) )
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Calculate the coordinates of the centroid of the original triangle: [ \text{Centroid} = \left( \frac{3 + 1 - 4}{3}, \frac{1 - 2 - 5}{3} \right) = \left( \frac{0}{3}, \frac{-6}{3} \right) = (0, -2) ]
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Dilate the coordinates of the centroid: Dilation factor: 5 Center of dilation: ( D(7, -6) )
[ \text{Dilated Centroid} = (7 + 5 \times 0, -6 + 5 \times (-2)) = (7, -16) ]
- Calculate the displacement of the centroid: [ \text{Displacement} = \text{Dilated Centroid} - \text{Original Centroid} ] [ \text{Displacement} = (7, -16) - (0, -2) = (7, -16) - (0, -2) = (7, -14) ]
So, the centroid will move 7 units to the right and 14 units upward.
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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.
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- 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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