A line segment goes from #(6 ,5 )# to #(7 ,3 )#. The line segment is dilated about #(2 ,1 )# by a factor of #3#. Then the line segment is reflected across the lines #x = 3# and #y=-4#, in that order. How far are the new endpoints form the origin?

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Answer 1 · 2018-04-12T03:10:20

Distance of the points from the origin after dilation and reflection :

#color(crimson)(bar(OA''') = 11.66, bar(OB''') = 20.62#

Given : #A(6,5), B(7,3)#

Now we have to find A', B' after rotation about point C (2,1) with a dilation factor of 3.

#A'(x, y) -> 3 * A(x,y) - C (x,y)#

#A'((x),(y)) = 3 * ((6), (5)) - ((2), (1)) = ((16),(14))#

Similarly, #B'((x),(y)) = 3 * ((7), (3)) - ((2), (1)) = ((19),(8))#

Now to find

Reflection Rules :

#color(crimson)("reflect over a line. ex: x=6, (2h-x, y)"#

#"Reflected across "x = 3#

#A'(x,y) -> A''(x,y) = ((6 - 16), (14)) = (-10, 14)#

#B'(x,y) -> B''(x,y) = ((2 * 3 - 19), (8)) = (-13,8)#

#"Reflected across " y = -4#

#color(crimson)("reflect over a line. ex: y= -3, (x, 2k-y)"#

#A''(x,y) -> A'''(x,y) = ((-10, 8-14) = (-10,6)#

#B''(x,y) -> B'''(x,y) =( -13, -8-8) = (-13, -16)#

#bar(OA''') = sqrt(-10^2 + 6^2) = 11.66#

#bar(OB''') = sqrt(-13^2 + -16^2) = 20.62#