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

Question

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

Aurora Ayala · Accepted Answer

New end points #color(purple)(A' (-1,0)#, #color(purple)(B' (11,-3)#

Length of the line segment after dilation is

#vec(A'B') ~~ color(green)(12.37)# rounded to 2 decimals

Given : A(1,4), B(5,3), Dilated around C(2,6), dilation factor 3 To find the new end points and length of line segment. #vec(A'C) = 3 * vec(AC)# or #a' = 3a - 2c# #a' = 3 ((1),(4)) - 2 ((2),(6)) = ((3),(12)) - ((4),(12)) = color(purple)(((-1),(0))# #vec(B'C) = 3 * vec(BC)# or #b' = 3b - 2c# #b' = 3 ((5),(3)) - 2 ((2),(6)) = ((15),(9)) - ((4),(12)) = color(purple)(((11),(-3))# #vec(A'B') = sqrt((-1-11)^2 + (0-(-3))^2) ~~ color(green)(12.37)# rounded to 2 decimals

Maverick Smith · Answer

undefined To dilate a line segment by a factor of $k$ around a point, we first find the new coordinates of the endpoints by applying the dilation formula. The formula for dilation around a point $(h, v)$ by a factor of $k$ is:

$$
(x', y') = (h + k(x - h), v + k(y - v))
$$

Given the endpoints (1, 4) and (5, 3), the point of dilation (center of dilation) is (2, 6), and the dilation factor is 3, we can calculate the new endpoints as follows:

For the first endpoint (1, 4):
$$
x' = 2 + 3(1 - 2) = -1
$$
$$
y' = 6 + 3(4 - 6) = 0
$$
So, the new endpoint is (-1, 0).

For the second endpoint (5, 3):
$$
x' = 2 + 3(5 - 2) = 11
$$
$$
y' = 6 + 3(3 - 6) = 3
$$
So, the new endpoint is (11, 3).

Therefore, the new endpoints of the line segment after dilation are (-1, 0) and (11, 3).

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

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

Substituting the coordinates of the new endpoints:

$$
\text{Length} = \sqrt{(11 - (-1))^2 + (3 - 0)^2} = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153}
$$

So, the length of the line segment after dilation is $ \sqrt{153} $ units.