# A line segment has endpoints at #(2 ,7 )# and #(5 ,4 )#. The line segment is dilated by a factor of #3 # around #(4 ,3 )#. What are the new endpoints and length of the line segment?

Please read the explanation.

Given:

Scale Factor for dilation is

Useful observations involving Dilation:

Isometry refers to a linear transformation which preserves the length.

Dilation is NOT an isometry. It creates similar figures only.

Here

The absolute value of the scale factor (k),

with the constraint

reduces the line segment

enlarges if otherwise.

Each point on the line segment

Dilation preserves the angle of measure.

Note that the pre-image and the image are parallel.

Observe that the points (center of dilation

And, the points (C, B and B') are also collinear.

Also, from

Move (4 x 3 = 12 units) up on the y-axis, and (2 x 3 = 6 units) left on the x-axis tor reach the end-point of

Similarly,

from

From

New end-points:

Find the length of

Hope this solution is helpful.

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To dilate a line segment by a factor of ( k ) around a given point, you need to follow these steps:

- Find the coordinates of the given point, which is (4, 3) in this case.
- Calculate the distance from the given point to each endpoint of the line segment.
- Multiply each distance by the dilation factor ( k ) to find the new distances.
- Use the new distances and the given point to find the coordinates of the new endpoints using the concept of similar triangles.

By following these steps, you can determine the new endpoints of the line segment after dilation. Additionally, you can calculate the length of the line segment using the distance formula with the coordinates of the new endpoints.

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

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