A line segment goes from $(5, 8)$ $(5 ,8 )$ to $(4, 2)$ $(4 ,2 )$ . The line segment is reflected across $x = - 3$ $x=-3$ , reflected across $y = - 5$ $y=-5$ , and then dilated about $(2, 0)$ $(2 ,0 )$ by a factor of $2$ $2$ . How far are the new endpoints from the origin?

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Answer 1 · 2017-06-05T22:53:56

$2 \sqrt{202} and 4 \sqrt{34}$ $2 sqrt 202 and 4 sqrt 34$

$A_{1} = (5, 8); B_{1} = (4, 2)$ $A_1 = (5, 8) ; B_1 = (4, 2)$

$- 3 + x \mapsto - 3 - x$ $-3 + x mapsto -3 - x$
$- 3 + x + 3 \mapsto - 3 - x - 3 = - 6 - x$ $-3 + x+3 mapsto -3 - x - 3 = -6 -x$

$A_{2} = (- 6 - 5, 8); B_{2} = (- 6 - 4, 2)$ $A_2 = (-6 -5, 8) ; B_2 = (-6 -4, 2)$

$- 5 + y \mapsto - 5 - y$ $-5 + y mapsto -5 - y$
$- 5 + y + 5 \mapsto - 5 - y - 5 = - 10 - y$ $-5 + y + 5 mapsto -5 - y - 5 = -10 -y$

$A_{3} = (- 11, - 10 - 8); B_{3} = (- 10, - 10 - 2)$ $A_3 = (-11, -10-8) ; B_3 = (-10, -10-2)$

What is to dilate the line about (2, 0) by a factor of 2?

I suppose $(x, y) \mapsto (2 x, y)$ $(x,y) mapsto (2x, y)$

$A_{4} = (- 11 \cdot 2, - 18); B_{4} = (- 10 \cdot 2, - 12)$ $A_4 = (-11 * 2, -18) ; B_4 = (-10 * 2, -12)$

${| A_{4} |}_{4}^{2} = 22^{2} + 18^{2}$ $|A_4|^2 = 22^2 + 18^2$

${| B_{4} |}_{4}^{2} = 20^{2} + 12^{2}$ $|B_4|^2 = 20^2 + 12^2$

A line segment goes from (5,8)(5 ,8 ) to (4,2)(4 ,2 ). The line segment is reflected across x=−3x=-3, reflected across y=−5y=-5, and then dilated about (2,0)(2 ,0 ) by a factor of 22. How far are the new endpoints from the origin?