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

Question

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

Maverick Smith · Accepted Answer

#(6,10),(8,0)# #"let "A(5,6) ,B(6,1)" and "D(4,2)# #" then "A'" and "B'" are the images of A and B under"# #"the dilatation"# #rArrvec(DA')=color(red)(2)vec(DA)# #rArrula'-uld=2(ula-uld)# #rArrula'-uld=2ula-2uld# #rArrula'=2ula-uld# #color(white)(xxxx)=2((5),(6))-((4),(2))# #color(white)(xxxx)=((10),(12))-((4),(2))=((6),(10))# #rArrA'(6,10)# #"similarly"# #vec(DB')=color(red)(2)vec(DB)# #rArrulb'-uld=2(ulb-uld)# #rArrulb'=2ulb-uld# #color(white)(xxxx)=2((6),(1))-((4),(2))=((8),(0))# #rArrB'(8,0)# #"to calculate length of segment use the "color(blue)"distance formula"# #•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# #"let "(x_1,y_1)=(6,10)" and "(x_2,y_2)=(8,0)# #d=sqrt((8-6)^2+(0-10)^2)=sqrt104~~10.2" 1 dec. place"#

Carson Arnold · Answer

undefined The new endpoints of the line segment after dilation by a factor of 2 around the point (4, 2) can be found using the dilation formula. Let's denote the original endpoints as $ A(5, 6) $ and $ B(6, 1) $.

1. Find the distance between the center of dilation (4, 2) and the original endpoints A and B.
$$ d_{A} = \sqrt{(5 - 4)^2 + (6 - 2)^2} = \sqrt{1^2 + 4^2} = \sqrt{17} $$
$$ d_{B} = \sqrt{(6 - 4)^2 + (1 - 2)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{5} $$

2. Multiply the distances $ d_{A} $ and $ d_{B} $ by the dilation factor, which is 2.
$$ d_{A_{new}} = 2 \times \sqrt{17} = 2\sqrt{17} $$
$$ d_{B_{new}} = 2 \times \sqrt{5} = 2\sqrt{5} $$

3. Find the coordinates of the new endpoints by extending the original segment by the respective dilated distances along the same line.
$$ A_{new} = (4, 2) + \frac{{(5, 6) - (4, 2)}}{{d_{A}}} \times d_{A_{new}} $$
$$ B_{new} = (4, 2) + \frac{{(6, 1) - (4, 2)}}{{d_{B}}} \times d_{B_{new}} $$

Solving these equations will give the new endpoints $ A_{new} $ and $ B_{new} $. Then you can calculate the length of the line segment using the distance formula between the new endpoints.