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

Mar 5, 2018

Distance of new points from origin

vec(OA”) = 33.94

vec(OB”) = 32.56

#### Explanation:

Given points $A \left(5 , 2\right) , B \left(4 , 2\right)$

Reflected across $x = - 3$, $y = - 5$

![Given points $A \left(5 , 2\right) , B \left(4 , 2\right)$

Reflected across $x = - 3$, $y = - 5$

A (x,y) -> A’(x, y) = (5,2) -> ((2x’ - 5), (2y’ - 2)) => (-11, -12)

A’(x,y) => ((-6 - 5), (-10 - 2)) => (-11, -12)

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

B’(x,y) => ((-6 - 4), (-10 - 2)) => (-10, -12)

New coordinates after reflection are A’(-11, -12), B’(-10, -12)

Now we we have to find A”, B” after rotation about point C (2,0) with a dilation factor of 2.

A”(x, y) -> 2 * A’(x,y) - C (x,y)

A”((x),(y)) = 2 * ((-11), (-12)) - ((2), (0)) = ((-24),(-24))

Similarly, B”((x),(y)) = 2 * ((-10), (-12)) - ((2), (0)) = ((-22),(-24))

New endpoints are A”(-24, -24), B”(-22, -24)

Distance of new points from origin

vec(OA”) = sqrt(-24^2 + -24^2) = 33.94

vec(OB”) = sqrt(-22^2 + -24^2) = 32.56]

A (x,y) -> A’(x, y) = (5,2) -> ((2x’ - 5), (2y’ - 2) => (-11, -12)

A’(x,y) => ((-6 - 5), (-10 - 2) => (-11, -12)

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

B’(x,y) => ((-6 - 4), (-10 - 2) => (-10, -12)

New coordinates after reflection are A’(-11, -12), B’(-10, -12)

Now we we have to find A”, B” after rotation about point C (2,0) with a dilation factor of 2.

A”(x, y) -> 2 * A’(x,y) - C (x,y)

A”((x),(y)) = 2 * ((-11), (-12)) - ((2), (0)) = ((-24),(-24))

Similarly, B”((x),(y)) = 2 * ((-10), (-12)) - ((2), (0)) = ((-22),(-24))

New endpoints are A”(-24, -24), B”(-22, -24)

Distance of new points from origin

vec(OA”) = sqrt(-24^2 + -24^2) = 33.94

vec(OB”) = sqrt(-22^2 + -24^2) = 32.56