# Point A is at #(4 ,1 )# and point B is at #(-6 ,-7 )#. Point A is rotated #(3pi)/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?

The new point A is at

The changed in distance between points A and B

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The new coordinates of point A after rotating clockwise about the origin by ( \frac{3\pi}{2} ) radians are (-1, 4). The distance between points A and B remains unchanged after the rotation. Therefore, the distance between the two points is the same as before, which can be calculated using the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Substituting the coordinates of points A and B into the formula:

[ d = \sqrt{(-6 - (-1))^2 + (-7 - 4)^2} = \sqrt{(-5)^2 + (-11)^2} = \sqrt{25 + 121} = \sqrt{146} ]

So, the distance between points A and B remains ( \sqrt{146} ) units.

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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 #(7 ,4 )# and #(3 ,5 )#. If the line segment is rotated about the origin by #(3 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 #(8 ,3 )# and #(1 ,4 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?
- A line segment has endpoints at #(2 , 3)# and #(1 , 2)#. If the line segment is rotated about the origin by #(pi)/2 #, translated vertically by #4#, and reflected about the x-axis, what will the line segment's new endpoints be?
- A triangle has corners at #(2 ,1 )#, #(4 ,-3 )#, and #(-1 ,4 )#. If the triangle is dilated by a factor of #5 # about point #(4 ,-9 ), how far will its centroid move?
- How does dilation affect the length of line segments?

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