A line segment has endpoints at #(5 ,9 )# and #(6 ,7 )#. The line segment is dilated by a factor of #4 # around #(5 ,1 )#. What are the new endpoints and length of the line segment?

1 Answer
marfre
Mar 9, 2017

#(5,33)# & #(9,24)#

Explanation:

Start by drawing the line in a rectangular coordinate plane and locating the point of dilation #(5,1)#.

Draw lines from #(5,1)# through each endpoint, extending above.

Since the point #(5,1)# and #(5,9)# have a #y#-difference of #8#. Multiply this by #4# to get #32#. Add #1# to get to the point location from #(5,1)#.

The slope of the original line = #-2/1#. From the point #(5,33)#, apply this slope #4# times to find the second endpoint. It needs to lie on the line from #(5,1)# through #(6,7)#.

You can see the process below:
enter image source here

The dilated line segment should be #4# times bigger.

Original line segment length #= sqrt(1^2 + 2^2) = sqrt(5) ~~ 2.236#

Dilated line segment length = #sqrt(8^2 + 4^2) = sqrt(80) = sqrt(16*5) = 4sqrt(5) ~~ 8.9443#

So endpoints of the dilated line are: #(5,33)# & #(9,24)#

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