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

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

1. Find the displacement vector from the center of dilation (3, 3) to each of the original endpoints (4, 3) and (2, 5).
2. Multiply each component of the displacement vector by the dilation factor (4).
3. Add the dilated displacement vector to the center of dilation to find the new endpoints.

Let's do the calculations:

For the endpoint (4, 3):

• Displacement vector: ( \begin{pmatrix} 4 - 3 \ 3 - 3 \end{pmatrix} = \begin{pmatrix} 1 \ 0 \end{pmatrix} )
• Dilated displacement vector: ( \begin{pmatrix} 4 \ 0 \end{pmatrix} )
• New endpoint: ( \begin{pmatrix} 3 \ 3 \end{pmatrix} + \begin{pmatrix} 4 \ 0 \end{pmatrix} = \begin{pmatrix} 7 \ 3 \end{pmatrix} )

For the endpoint (2, 5):

• Displacement vector: ( \begin{pmatrix} 2 - 3 \ 5 - 3 \end{pmatrix} = \begin{pmatrix} -1 \ 2 \end{pmatrix} )
• Dilated displacement vector: ( \begin{pmatrix} -4 \ 8 \end{pmatrix} )
• New endpoint: ( \begin{pmatrix} 3 \ 3 \end{pmatrix} + \begin{pmatrix} -4 \ 8 \end{pmatrix} = \begin{pmatrix} -1 \ 11 \end{pmatrix} )

So, the new endpoints are (7, 3) and (-1, 11).

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

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

Using the new endpoints (7, 3) and (-1, 11):

[ \text{Length} = \sqrt{(-1 - 7)^2 + (11 - 3)^2} ] [ \text{Length} = \sqrt{(-8)^2 + (8)^2} ] [ \text{Length} = \sqrt{64 + 64} ] [ \text{Length} = \sqrt{128} ] [ \text{Length} = 8\sqrt{2} ]

Therefore, the new length of the line segment is ( 8\sqrt{2} ).