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

##### 1 Answer
Jul 7, 2016

Original segment ${A}_{0} {B}_{0}$, where ${A}_{0} = \left(1 , 2\right) , {B}_{0} = \left(4 , 7\right)$,
is transformed into $A B$, where $A = \left(21 , - 9\right) , B = \left(15 , - 19\right)$.
The distances from the origin to the new endpoints are
${d}_{A} \approx 22.8$
${d}_{B} \approx 24.2$

#### Explanation:

1. Reflection of a point with coordinates $\left({a}_{0} , {b}_{0}\right)$ relative to a line $x = 6$ (vertical line intersecting X-axis at coordinate $x = 6$) will be horizontally shifted into a new X-coordinate obtained by adding to an X-coordinate of the axis of symmetry ($x = 6$) the distance from it of the original X-coordinates ($6 - {a}_{0}$).
Y-coordinate remains the same in this transformation.
So, new coordinates are:
$\left({a}_{1} , {b}_{1}\right) = \left(6 + \left(6 - {a}_{0}\right) , {b}_{0}\right) = \left(12 - {a}_{0} , {b}_{0}\right)$

2. Reflection of a point with coordinates $\left({a}_{1} , {b}_{1}\right)$ relative to a line $y = - 1$ (horizontal line intersecting Y-axis at coordinate $y = - 1$) will be vertically shifted into a new Y-coordinate obtained by adding to an Y-coordinate of the axis of symmetry ($y = - 1$) the distance from it of the original Y-coordinates ($- 1 - {b}_{1}$).
X-coordinate remains the same in this transformation.
So, new coordinates are:
$\left({a}_{2} , {b}_{2}\right) = \left({a}_{1} , - 1 + \left(- 1 - {b}_{1}\right)\right) =$
$= \left({a}_{1} , - 2 - {b}_{1}\right) = \left(12 - {a}_{0} , - 2 - {b}_{0}\right)$

3. Dilation about a center point $\left(1 , 1\right)$ by a factor of $2$ will transform a point $\left({a}_{2} , {b}_{2}\right)$ into
$\left({a}_{3} , {b}_{3}\right) = \left(1 + 2 \left({a}_{2} - 1\right) , 1 + 2 \left({b}_{2} - 1\right)\right) =$
$= \left(1 + 2 \left(12 - {a}_{0} - 1\right) , 1 + 2 \left(- 2 - {b}_{0} - 1\right)\right) =$
$= \left(23 - 2 {a}_{0} , - 5 - 2 {b}_{0}\right)$

4. Using this formula for both ends of our original segment $A B$, where $A \left(1 , 2\right)$ and $B = \left(4 , 7\right)$:
4.1. $\left({a}_{0} = 1 , {b}_{0} = 2\right)$
$\rightarrow$ $\left({a}_{3} = 23 - 2 \cdot 1 , {b}_{3} = - 5 - 2 \cdot 2\right) =$
$= \left(21 , - 9\right)$
4.2. $\left({a}_{0} = 4 , {b}_{0} = 7\right)$
$\rightarrow$ $\left({a}_{3} = 23 - 2 \cdot 4 , {b}_{3} = - 5 - 2 \cdot 7\right) =$
$= \left(15 , - 19\right)$

5. The distance of each end of a new segment from the origin are
${d}_{A} = \sqrt{{\left(21\right)}^{2} + {\left(- 9\right)}^{2}} = \sqrt{552} \approx 22.8$
${d}_{B} = \sqrt{{\left(15\right)}^{2} + {\left(- 19\right)}^{2}} = \sqrt{586} \approx 24.2$