# Point A is at #(-7 ,7 )# and point B is at #(5 ,-3 )#. 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' = (7,-7) , d ≈ 10.85

a point A (x,y) → A' (-x,-y)

hence A (-7,7) → A' (7,-7)

To calculate the change in distance we will need to calculate the distance from A to B and also the distance from A' to B.

distance between A and B

distance between A' and B

change in distance = 15.62 - 4.77 = 10.85

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To find the new coordinates of point A after rotating it π radians clockwise about the origin, we use the following rotation formula for a point (x, y):

[ (x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta)) ]

Where ( (x', y') ) are the new coordinates, ( (x, y) ) are the original coordinates, and ( \theta ) is the angle of rotation.

Given that point A is at (-7, 7) and we're rotating it by π radians clockwise, we have:

[ x = -7, , y = 7, , \theta = \pi ]

Plugging these values into the rotation formula:

[ x' = -7 \cos(\pi) - 7 \sin(\pi) ] [ y' = -7 \sin(\pi) + 7 \cos(\pi) ]

[ x' = -7(-1) - 7(0) = 7 ] [ y' = -7(0) + 7(-1) = -7 ]

So, the new coordinates of point A are (7, -7).

To find the change in distance between points A and B, we calculate the distances before and after the rotation.

The distance formula between two points (x1, y1) and (x2, y2) is given by:

[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} ]

For points A and B, the initial distance is:

[ d_{AB} = \sqrt{(5 - (-7))^2 + (-3 - 7)^2} = \sqrt{(12)^2 + (-10)^2} = \sqrt{144 + 100} = \sqrt{244} ]

After rotating point A, the new distance between the rotated point A' and point B remains the same, as rotation about the origin doesn't affect the distance between two points. Therefore, the change in distance between points A and B is zero.

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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.

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