A line segment goes from $(3 ,4 )$ to $(5 ,1 )$ . The line segment is dilated about $(1 ,0 )$ by a factor of $2$ . Then the line segment is reflected across the lines $x=-2$ and $y=2$ , in that order. How far are the new endpoints from the origin?

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Answer 1 · 2018-04-21T11:39:19

If I've done it right the first is $sqrt{85}$ from the origin and the second is $11$ from the origin.

Thought I just did one of these.

Shifting the dilation point to the origin maps the other points:

$(3,4)-(1,0)=(2,4) quad quad quad quad (5,1)-(1,0)=(4,1)$

Doubling each dilates around the origin by a factor of two.

$2(2,4) = (4,8) quad quad quad quad 2(4,1)=(8,2)$

Now we shift the origin back to the dilation point:

$(4,8)+(1,0)=(5,8) quad quad quad quad (8,2)+(1,0)=(9,2)$

Now we reflect through $x=-2$ which leaves the $y$ coordinate alone.

$(-2 - 5,8)=(-7,8) quad quad quad quad (-2-9,2)=(-11,2)$

Now we reflect through $y=2$ which leaves the $x$ coordinate alone.

$(-7,2-8)=(-7,-6) quad quad quad quad (-11,2-2)=(-11,0)$

If I've done that right the first is $sqrt{7^2+6^2}=sqrt{85}$ from the origin and the second is $11$ from the origin.

A line segment goes from (3 ,4 )(3 ,4 ) to (5 ,1 )(5 ,1 ). The line segment is dilated about (1 ,0 )(1 ,0 ) by a factor of 22. Then the line segment is reflected across the lines x=-2x=-2 and y=2y=2, in that order. How far are the new endpoints from the origin?