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?
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To dilate a line segment by a factor of ( k ) around a point, we need to:
 Translate the line segment so that the center of dilation coincides with the origin.
 Dilate the translated line segment by multiplying the coordinates of its endpoints by ( k ).
 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:

Translate the line segment so that ( C ) coincides with the origin:
 Subtract the coordinates of ( C ) from the coordinates of ( A ) and ( B ): ( A'(55, 93) = A'(0, 6) ) and ( B'(85, 23) = B'(3, 1) ).

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) ).

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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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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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