A line segment has endpoints at #(3 ,2 )# and #(5 ,4 )#. The line segment is dilated by a factor of #4 # around #(1 ,3 )#. What are the new endpoints and length of the line segment?

Answer 1

#color(brown)("New coordinates are " (9, -1), (17, 7)#

#color(brown)("Length of the line segment '" = sqrt((9-17)^2 + (-1-7)^2) ~~ 11.3137#

#A(8,4), B(3,2), " about point " D (1,3), " dilation factor "4#
#A'((x),(y)) =(4)a - (3)d =(4)*((3),(2)) - (3)*((1),(3)) = ((9),(-1))#
#B'((x),(y)) = (4)b - (3)d = (4)*((5),(4)) - (3)*((1),(3)) = ((17),(7)) #
#color(brown)("New coordinates are " (9, -1), (17, 7)#
#color(brown)("Length of the line segment '" = sqrt((9-17)^2 + (-1-7)^2) ~~ 11.3137#
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Answer 2

To find the new endpoints after dilation, we first calculate the distance between the center of dilation and each endpoint. Then, we multiply these distances by the dilation factor (4) to find the new distances. Finally, we add these new distances to the coordinates of the center of dilation to find the new endpoints.

Center of dilation: (1, 3)

Endpoint 1: (3, 2) Distance from center of dilation: √((1 - 3)^2 + (3 - 2)^2) = √((-2)^2 + (1)^2) = √(4 + 1) = √5 New distance: 4 * √5 = 4√5 New endpoint 1: (1 + 4√5, 3 + 4√5) ≈ (7.47, 7.47)

Endpoint 2: (5, 4) Distance from center of dilation: √((1 - 5)^2 + (3 - 4)^2) = √((-4)^2 + (-1)^2) = √(16 + 1) = √17 New distance: 4 * √17 = 4√17 New endpoint 2: (1 + 4√17, 3 + 4√17) ≈ (17.71, 17.71)

New endpoints: (7.47, 7.47) and (17.71, 17.71)

To find the length of the line segment, we use the distance formula between the new endpoints:

Length = √((17.71 - 7.47)^2 + (17.71 - 7.47)^2) Length ≈ √((10.24)^2 + (10.24)^2) Length ≈ √(104.86 + 104.86) Length ≈ √209.72 Length ≈ 14.49

The length of the dilated line segment is approximately 14.49 units.

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Answer from HIX Tutor

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.

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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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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