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

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Answer 1 · 2018-04-12T07:58:36

$color(blue)("Distances of A & B from origin after dilation and reflection " 9.49, 13.04 " respectively"$

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

$A(x,y) -> A'(x,y) = 2 * A(x,y) - C(x,y) = (2*(3,2)-(1,1)) = (5, 3)$

$B(x,y) -> B'(x,y) = 2 * B(x,y) - C(x,y) = (2*(1,3)-(1,1)) = (1, 5)$

Line segment A'B' reflected across x = 4, y = -3, in that order.

$color(crimson)("reflect thru x = 4, y = -3 ", h=4, k= -3. (2h-x, 2k-y)"$

$A''(x, y) = (2h - x, 2k-y) = (2*4 - 5, 2*-3 - 3) = (3, -9)$

$B''(x, y) = (2h - x, 2k-y) = (2*4 - 1, 2*-3 - 5) = (7, -11)$

$OA'' = sqrt(3^2 + 9^2) = 9.49$

$OB'' = sqrt(7^2 + 11^2) = 13.04$

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