A line segment goes from (1 ,5 ) to (7 ,6 ). The line segment is dilated about (1 ,2 ) by a factor of 3. Then the line segment is reflected across the lines x = 4 and y=-2, in that order. How far are the new endpoints form the origin?

1 Answer
May 30, 2018

new endpoints: $A \left(- 11 , - 18\right) , B \left(7 , - 15\right)$
d_A = sqrt(445) ~~21.1; d_B = sqrt(274)~~16.6

Explanation:

Given: $A \left(1 , 5\right) , B \left(7 , 6\right)$. Line $\overline{A B}$ is dilated about $\left(1 , 2\right)$ by a factor $k = 3$. Then the line segment is reflected across $x = 4$ and $y = - 2$.

Dilated about $\left(1 , 2\right)$ . Find the distance to each endpoint from the dilated point:

${d}_{A \to \left(1 , 2\right)} = \sqrt{{\left(5 - 2\right)}^{2} + {\left(1 - 1\right)}^{2}} = \sqrt{9} = 3$

${d}_{B \to \left(1 , 2\right)} = \sqrt{{\left(6 - 2\right)}^{2} + {\left(7 - 1\right)}^{2}} = \sqrt{16 + 36} = \sqrt{52} = 2 \sqrt{13}$

Triple the distance (dilate the length to the point (1, 2)):
3(3) = 9; " " 3*sqrt(52) = 3sqrt(52) = 6sqrt(13)

Since $\left(1 , 2\right)$ & $\left(1 , 5\right)$ have the same $x$ value, the dilated point has the same $x$ value:

$\text{ } A ' \left(1 , 2 + 9\right) = A ' \left(1 , 11\right)$
The 2 is added from the dilated point's $y$ value.

Use proportions to find the $B ' \left({x}_{B} , {y}_{B}\right)$ coordinates:

(2sqrt(13))/(6sqrt(13)) = 6/x; " "2sqrt(13)x = 36 sqrt(13); " "x = (36 sqrt(13))/(2 sqrt(13)) = 18

${x}_{B} = 1 + 18 = 19$
The 1 is added from the dilated point's $x$ value.

(2sqrt(13))/(6sqrt(13)) = 4/y; " "2sqrt(13)y = 24 sqrt(13); " "y = (24 sqrt(13))/(2 sqrt(13)) = 12

${y}_{B} = 2 + 12 = 14$
The 2 is added from the dilated point's $y$ value.

Dilated line segment points: $\text{ "A'(1, 11); " } B ' \left(19 , 14\right)$

Reflected about the $x = 4$ line:

$x$ distance from $x = 4$ to ${x}_{A '} = 3$
$x$ distance from $x = 4$ to ${x}_{B '} = 15$

Since the points are on opposite sides of $x = 4$ there are two coordinate rules:

coordinate rule of reflected point A' :$\left(x , y\right) \to \left(4 + x \text{-distance} , y\right)$
coordinate rule of reflected point B' :$\left(x , y\right) \to \left(4 - x \text{-distance} , y\right)$

$A ' ' \left(4 + 3 , 11\right) = \left(7 , 11\right)$
$B ' ' \left(4 - 15 , 14\right) = \left(- 11 , 14\right)$

Reflected about $x = 4$ line segment points: A'(7, 11); " "B'(-11, 14)

Reflected about the $y = - 2$ line:

$y$ distance from $y = 2$ to ${y}_{A ' '} = 13$
$y$ distance from $y = 2$ to ${y}_{B ' '} = 16$

coordinate rule of reflected points :$\left(x , y\right) \to \left(x , - 2 - y \text{-distance}\right)$

$A ' ' ' \left(7 , - 2 - 13\right) = \left(7 , - 15\right)$
$B ' ' ' \left(- 11 , - 2 - 16\right) = \left(- 11 , - 18\right)$

Reflected about $y = - 2$ line segment points: A'(7, -15); " "B'(-11, -18)

distance of each endpoint from the origin:

${d}_{A ' ' ' \to \left(0 , 0\right)} = \sqrt{{7}^{2} + {\left(- 15\right)}^{2}} = \sqrt{274} \approx 16.6$

${d}_{B ' ' ' \to \left(0 , 0\right)} = \sqrt{{\left(- 11\right)}^{2} + {\left(- 18\right)}^{2}} = \sqrt{445} \approx 21.1$