A line segment goes from $(2 ,6 )$ to $(1 ,3 )$ . The line segment is dilated about $(2 ,0 )$ by a factor of $2$ . Then the line segment is reflected across the lines $x = 2$ and $y=5$ , in that order. How far are the new endpoints from the origin?

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Answer 1 · 2018-04-12T04:04:22

$color(green)("Dist of A from origin after dilation and reflections " = 2.83$

$color(green)("Dist of B from origin after dilation and reflections " = 5.66$

$A(2,6), B(1,3), " Dilated about " C(2,0) " by factor " 3$

$A'(x,y) -> 2 * A(x,y) - C(x,y) = ((4,12) - (2,0)) = (2,12)$

$B'(x,y) -> 2 * B(x,y) - C(x,y) = ((2,6) - (2,0)) = (0,6)$

Line segment A'B' reflected across x = 2, y = 5 in that order.

Reflection Rule :

$color(crimson)("reflect over a line. ex: x=h. (2h-x, y)"$

#A'(x,y) -> A''(x,y) = (4 - 2, 12) -> (2,12)

$B'(x,y) - > B''(x,y) = (4-0, 6) -> (4,6)$

$color(crimson)("reflect over a line. ex: y= k. (x, 2k-y)"$

$A''(x,y) -> A'''(x,y) = (2, 10-12) -> (2,-2)$

$B''(x,y) -> B'''(x,y) = (4, 10-6) -> (4,4)$

$bar(A'''O) = sqrt(2^2 + 2^2) = 2.83$

$bar(B'''O) = sqrt(4^2 + 4^2) = 5.66$

#color(green)("Dist of A from origin after dilation and reflections " = 2.83#

$color(green)("Dist of B from origin after dilation and reflections " = 5.66$

A line segment goes from (2 ,6 )(2 ,6 ) to (1 ,3 )(1 ,3 ). The line segment is dilated about (2 ,0 )(2 ,0 ) by a factor of 22. Then the line segment is reflected across the lines x = 2x = 2 and y=5y=5, in that order. How far are the new endpoints from the origin?