# A triangle has corners at #(8 ,3 )#, #(4 ,-6 )#, and #(-2 ,-4 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-3 ), how far will its centroid move?

Distance moved by centroid

To find the distance moved by centroid.

Similarly,

New centroid after dilation

Distance moved by centroid

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To find the centroid of the triangle, first, calculate the coordinates of the centroid of the original triangle. Then, dilate the coordinates of the centroid using the same scale factor.

Centroid of the original triangle: [ x_{\text{centroid}} = \frac{x_1 + x_2 + x_3}{3} ] [ y_{\text{centroid}} = \frac{y_1 + y_2 + y_3}{3} ]

Coordinates of the original triangle: ( (8, 3), (4, -6), (-2, -4) )

Using the formula: [ x_{\text{centroid}} = \frac{8 + 4 - 2}{3} = \frac{10}{3} ] [ y_{\text{centroid}} = \frac{3 - 6 - 4}{3} = -\frac{7}{3} ]

Now, dilate the centroid using the scale factor of 5 about point (1, -3):
[ x'*{\text{centroid}} = x*{\text{center}} + 5 \times (x_{\text{centroid}} - x_{\text{center}}) ]
[ y'*{\text{centroid}} = y*{\text{center}} + 5 \times (y_{\text{centroid}} - y_{\text{center}}) ]

Given the center point as (1, -3):
[ x'*{\text{centroid}} = 1 + 5 \times \left(\frac{10}{3} - 1\right) = \frac{25}{3} ]
[ y'*{\text{centroid}} = -3 + 5 \times \left(-\frac{7}{3} + 3\right) = -\frac{16}{3} ]

The coordinates of the centroid of the dilated triangle are ( \left(\frac{25}{3}, -\frac{16}{3}\right) ).

Now, calculate how far the centroid has moved from the original centroid:
[ \text{Distance} = \sqrt{(x'*{\text{centroid}} - x*{\text{centroid}})^2 + (y'*{\text{centroid}} - y*{\text{centroid}})^2} ]

Substitute the values: [ \text{Distance} = \sqrt{\left(\frac{25}{3} - \frac{10}{3}\right)^2 + \left(-\frac{16}{3} + \frac{7}{3}\right)^2} ] [ \text{Distance} = \sqrt{\left(\frac{15}{3}\right)^2 + \left(-\frac{9}{3}\right)^2} ] [ \text{Distance} = \sqrt{\left(\frac{225}{9}\right) + \left(\frac{81}{9}\right)} ] [ \text{Distance} = \sqrt{\frac{306}{9}} ] [ \text{Distance} = \sqrt{\frac{102}{3}} ] [ \text{Distance} = \sqrt{\frac{102}{3}} ] [ \text{Distance} = \frac{\sqrt{102}}{\sqrt{3}} ] [ \text{Distance} = \frac{\sqrt{102}}{3} ]

So, the centroid of the dilated triangle moves ( \frac{\sqrt{102}}{3} ) units away from the original centroid.

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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 triangle has corners at #(-2 ,3 )#, #(3 ,2 )#, and #(5 ,-6 )#. If the triangle is dilated by a factor of #2 # about point #(1 ,-8 ), how far will its centroid move?
- Point A is at #(-7 ,3 )# and point B is at #(5 ,4 )#. Point A is rotated #pi # 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?
- A line segment has endpoints at #(0 ,2 )# and #(3 ,5 )#. If the line segment is rotated about the origin by #( 3 pi)/2 #, translated vertically by # 3 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- A line segment with endpoints at #(1 , -2 )# and #(6, 9 )# is rotated clockwise by #(3 pi)/2#. What are the new endpoints of the line segment?
- Point A is at #(-7 ,7 )# and point B is at #(5 ,1 )#. Point A is rotated #pi # 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?

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