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

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Answer 1 · 2018-08-07T02:17:58

$color(purple)("Distances of A & B after dilation and reflection "$

$color(green)(sqrt 85, sqrt73 " respectivelyy."$

$A (2, 3), B (4,1), " dilated by factor 3 about " C(0,1)$

$A(x,y) -> A'(x,y) = 3* A(x,y) - 2*C(x,y) = (3*(2,3) - 2*(0,1)) = (6,7)$

$B(x,y) -> B'(x,y) = 3* B(x,y) - 2*C(x,y) = (3*(4,1) - 2*(0,1)) = (12,1)$

$"Reflection Rule : reflect thru " x = 2, y = -1; h=2, k= -1; (2h-x, 2k-y)$

$A''(x,y) = A'((2h - x), (2k - y)) = (4-6, -2-7) = (-2, -9)$

$B''(x,y) = B'((2h - x), (2k - y)) = (4-12, -2-1) = (-8, -3)$

$OA'' = sqrt(-2^2 + -9^2) = sqrt85$

$OB'' = sqrt(-8^2 + -3^2) = sqrt 73$

A line segment goes from (2 ,3 )(2 ,3 ) to (4 ,1 )(4 ,1 ). The line segment is dilated about (0 ,1 )(0 ,1 ) by a factor of 33. Then the line segment is reflected across the lines x=2x=2 and y=-1y=-1, in that order. How far are the new endpoints from the origin?