# A line segment has endpoints at #(5 ,8 )# and #(2 , 1)#. The line segment is dilated by a factor of #3 # around #(6 , 4)#. What are the new endpoints and length of the line segment?

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To find the new endpoints of the line segment after dilation by a factor of 3 around the point (6, 4), follow these steps:

- Find the vector representing the original line segment by subtracting the coordinates of one endpoint from the coordinates of the other endpoint.
- Multiply this vector by the dilation factor (3) to obtain the vector representing the dilated line segment.
- Add the coordinates of the center of dilation (6, 4) to this dilated vector to find the coordinates of the new endpoints.

Let's go through these steps:

Original line segment: Endpoint 1: (5, 8) Endpoint 2: (2, 1)

Vector representing the original line segment: v = (2 - 5, 1 - 8) = (-3, -7)

Dilated vector: v_dilated = (3 * -3, 3 * -7) = (-9, -21)

New endpoints: Endpoint 1: (6, 4) + (-9, -21) = (-3, -17) Endpoint 2: (6, 4) + (-9, -21) = (-3, -17)

So, the new endpoints of the line segment after dilation are (-3, -17) and (-3, -17).

To find the length of the dilated line segment, you can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

For the new endpoints (-3, -17) and (-3, -17):

Length = √((-3 - (-3))^2 + (-17 - (-17))^2) = √(0^2 + 0^2) = √(0) = 0

Therefore, the length of the dilated line segment is 0.

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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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