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

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

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Answer 1 · 2018-04-12T02:24:31

$¯ ¯¯¯¯ ¯ O A \approx 14.87$ $color(crimson)( bar(OA") ~~ 14.87$

$¯ ¯¯¯¯ ¯ O B \approx 16.12$ $color(crimson)( bar(OB") ~~ 16.12$

Explanation:

Given points $A (1, 2), B (4, 1)$ $A(1,2) , B (4,1)$

Reflected across $x = - 1$ $x = -1$ , $y = 3$ $y = 3$

$Reflection Rules :$ $color(blue)("Reflection Rules :"$
$reflect over x-axis. (x,-y)$ $color(blue)("reflect over x-axis. (x,-y)"$
$reflect over y-axis. (-x,y)$ $color(blue)("reflect over y-axis. (-x,y)"$
$reflect over line y=x. (y,x)$ $color(blue)("reflect over line y=x. (y,x)"$
$reflect over line y= -x. (-y,-x)$ $color(blue)("reflect over line y= -x. (-y,-x)"$
$reflect thru origin. (-x,-y)$ $color(blue)("reflect thru origin. (-x,-y)"$

$reflect thru a different point. ex: (5,-1) h=5 k= -1. (2h-x, 2k-y)$ $color(brown)("reflect thru a different point. ex: (5,-1) h=5 k= -1. (2h-x, 2k-y)"$

$reflect over a line. ex: x=6. (2h-x, y)$ $color(blue)("reflect over a line. ex: x=6. (2h-x, y)"$
$reflect over a line. ex: y= -3. (x, 2k-y)$ $color(blue)("reflect over a line. ex: y= -3. (x, 2k-y)"$

![https://www.onlinemath4all.com/http://reflection-transformation.html](https://useruploads.socratic.org/jnL1j2U4SgalH5rQVgKp_82D26EA3-2700-4F92-8282-C0AD796BB9EE.png)

$A (x,y) -> A’(x, y) = (1,2) -> ((2x’- 1), (2y’ - 2))$

$A’(x,y) => ((-2 - 1), (6 - 2)) => (-3, 4)$

$B (x,y) - > B’ (x,y) = (4 , 1) -> ((2x’ - 4), (2y’ - 1)$

$B’(x,y) => ((-2 - 4), (6 - 1) => (-6, 5)$

New coordinates after reflection are $A’(-3, 4), B’(-6, 5)$

Now we we have to find A”, B” after rotation about point C (2,2) with a dilation factor of 3.

$A”(x, y) -> 3 * A’(x,y) - C (x,y)$

$A”((x),(y)) = 3 * ((-3), (4)) - ((2), (2)) = ((-11),(10))$

Similarly, $B”((x),(y)) = 2 * ((-6), (5)) - ((2), (2)) = ((-14),(8))$

New endpoints are $A”(-11, 10), B”(-14, 8)$

Distance of new points from origin

$vec(OA”) = sqrt(-11^2 + 10^2) ~~ 14.87$

$vec(OB”) = sqrt(-14^2 + 8^2) ~~ 16.12$

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

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Explanation:

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A line segment goes from (1 ,2 )(1,2)(1 ,2 ) to (4 ,1 )(4,1)(4 ,1 ). The line segment is reflected across x=-1x=−1x=-1, reflected across y=3y=3y=3, and then dilated about (2 ,2 )(2,2)(2 ,2 ) by a factor of 333. How far are the new endpoints from the origin?

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Explanation:

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