A line segment has endpoints at #(5 ,9 )# and #(8 ,2 )#. The line segment is dilated by a factor of #2 # around #(5 ,3 )#. What are the new endpoints and length of the line segment?

Answer 1

#color(cyan)("New end points are " (5,15), (11,1)#

#color(cyan)("Length of the line segment '" ~~ 15.2315#

#A(5,9), B(8,2), " about point " D (5,3), " dilation factor " 2#
#A'((x),(y)) =(2)a - (1)d =(2)* ((5),(9)) - (1)*((5),(3)) = ((5),(15))#
#B'((x),(y)) = (2)b - (1)d = (2)* ((8),(2)) - (1)*((5),(3)) = ((11),(1)#
#color(cyan)("New end points are " (5,15), (11,1)#
#color(cyan)("Length of the line segment '" = sqrt((5-11)^2 + (15-1)^2) ~~ 15.2315#
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Answer 2

To dilate a line segment by a factor of ( k ) around a point, we need to:

  1. Translate the line segment so that the center of dilation coincides with the origin.
  2. Dilate the translated line segment by multiplying the coordinates of its endpoints by ( k ).
  3. Translate the dilated line segment back to its original position.

Given endpoints ( A(5, 9) ) and ( B(8, 2) ) and a dilation factor of ( k = 2 ) around the point ( C(5, 3) ), we follow these steps:

  1. Translate the line segment so that ( C ) coincides with the origin:

    • Subtract the coordinates of ( C ) from the coordinates of ( A ) and ( B ): ( A'(5-5, 9-3) = A'(0, 6) ) and ( B'(8-5, 2-3) = B'(3, -1) ).
  2. Dilate the translated line segment by multiplying the coordinates of its endpoints by ( k = 2 ):

    • ( A''(2 \times 0, 2 \times 6) = A''(0, 12) ) and ( B''(2 \times 3, 2 \times -1) = B''(6, -2) ).
  3. Translate the dilated line segment back to its original position:

    • Add the coordinates of ( C ) to the coordinates of ( A'' ) and ( B'' ): ( A'''(0+5, 12+3) = A'''(5, 15) ) and ( B'''(6+5, -2+3) = B'''(11, 1) ).

Therefore, the new endpoints of the line segment after dilation are ( A'''(5, 15) ) and ( B'''(11, 1) ).

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 endpoints ( A'''(5, 15) ) and ( B'''(11, 1) ):

[ \text{Length} = \sqrt{(11 - 5)^2 + (1 - 15)^2} ] [ = \sqrt{6^2 + (-14)^2} ] [ = \sqrt{36 + 196} ] [ = \sqrt{232} ]

So, the length of the line segment after dilation is ( \sqrt{232} ).

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Answer from HIX Tutor

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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