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

##### 1 Answer
Apr 12, 2018

color(crimson)( bar(OA") ~~ 14.87

color(crimson)( bar(OB") ~~ 16.12

#### Explanation:

Given points $A \left(1 , 2\right) , B \left(4 , 1\right)$

Reflected across $x = - 1$, $y = 3$

color(blue)("Reflection Rules :"
color(blue)("reflect over x-axis. (x,-y)"
color(blue)("reflect over y-axis. (-x,y)"
color(blue)("reflect over line y=x. (y,x)"
color(blue)("reflect over line y= -x. (-y,-x)"
color(blue)("reflect thru origin. (-x,-y)"

color(brown)("reflect thru a different point. ex: (5,-1) h=5 k= -1. (2h-x, 2k-y)"

color(blue)("reflect over a line. ex: x=6. (2h-x, y)"
color(blue)("reflect over a line. ex: y= -3. (x, 2k-y)"

A (x,y) -> A’(x, y) = (1,2) -> ((2x’- 1), (2y’ - 2))

A’(x,y) => ((-2 - 1), (6 - 2)) => (-3, 4)

B (x,y) - > B’ (x,y) = (4 , 1) -> ((2x’ - 4), (2y’ - 1)

B’(x,y) => ((-2 - 4), (6 - 1) => (-6, 5)

New coordinates after reflection are A’(-3, 4), B’(-6, 5)

Now we we have to find A”, B” after rotation about point C (2,2) with a dilation factor of 3.

A”(x, y) -> 3 * A’(x,y) - C (x,y)

A”((x),(y)) = 3 * ((-3), (4)) - ((2), (2)) = ((-11),(10))

Similarly, B”((x),(y)) = 2 * ((-6), (5)) - ((2), (2)) = ((-14),(8))

New endpoints are A”(-11, 10), B”(-14, 8)

Distance of new points from origin

vec(OA”) = sqrt(-11^2 + 10^2) ~~ 14.87

vec(OB”) = sqrt(-14^2 + 8^2) ~~ 16.12