A line segment has endpoints at #(9 ,2 )# and #(5 , 4)#. The line segment is dilated by a factor of #3 # around #(2 , 3)#. What are the new endpoints and length of the line segment?

1 Answer
Jake M.
Mar 22, 2018

(23, 0) and (11, 6), #6 sqrt5#

Explanation:

Dilating around a point is hard unless it's the origin. So we can shift to the origin and then shift back.

So we subtract (2, 3) from (9,2) and (5, 4) and get (7, -1) and (3, 1). We now dilate them by 3, yielding (21, -3) and (9, 3). We finally shift them back by adding (2, 3) and get (23, 0) and (11, 6).

From that, we have the length of the line segment is
#L = sqrt(Delta x^2 + Delta y^2) = sqrt((11 - 23)^2 + (6 - 0)^2) #
#L = sqrt(144 + 36) =6 sqrt(5) #

For the second step, we could also have gotten the original length and multiplied by the dilation factor, i.e.
#L_0 = sqrt((5-9)^2 + (4 - 2)^2) = sqrt(16 + 4) = 2sqrt(5)#
#L = 3 * L_0 = 6 sqrt(5) #

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