# 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?

Apr 12, 2018

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

#### Explanation:

$A \left(3 , 2\right) , B \left(1 , 3\right) , \text{ dilated by 2 about C(1,1)}$

$A \left(x , y\right) \to A ' \left(x , y\right) = 2 \cdot A \left(x , y\right) - C \left(x , y\right) = \left(2 \cdot \left(3 , 2\right) - \left(1 , 1\right)\right) = \left(5 , 3\right)$

$B \left(x , y\right) \to B ' \left(x , y\right) = 2 \cdot B \left(x , y\right) - C \left(x , y\right) = \left(2 \cdot \left(1 , 3\right) - \left(1 , 1\right)\right) = \left(1 , 5\right)$

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)"

$A ' ' \left(x , y\right) = \left(2 h - x , 2 k - y\right) = \left(2 \cdot 4 - 5 , 2 \cdot - 3 - 3\right) = \left(3 , - 9\right)$

$B ' ' \left(x , y\right) = \left(2 h - x , 2 k - y\right) = \left(2 \cdot 4 - 1 , 2 \cdot - 3 - 5\right) = \left(7 , - 11\right)$

$O A ' ' = \sqrt{{3}^{2} + {9}^{2}} = 9.49$

$O B ' ' = \sqrt{{7}^{2} + {11}^{2}} = 13.04$