# A line segment has endpoints at (5 , 2) and (3 , 1). If the line segment is rotated about the origin by pi/2 , translated vertically by -2, and reflected about the y-axis, what will the line segment's new endpoints be?

May 24, 2018

color(blue)((-2,-7)

color(blue)((-1,-5)

#### Explanation:

No direction of rotation is given, so I will take this as anti-clockwise:

A rotation of $\frac{\pi}{2}$ anti-clockwise maps:

$\left(x , y\right) \to \left(y , - x\right)$

A translation of -2 units map:

$\left(x , y\right) \to \left(x , y - 2\right)$

A reflection in the y axis map:

$\left(x , y\right) \to \left(- x , y\right)$

If we combine these, we get:

$\left(x , y\right) \to \left(y , - x\right) \to \left(y , - x - 2\right) \to \left(- y , - x - 2\right)$

Therefore:

$\left(5 , 2\right) \to \left(- y , - x - 2\right) = \left(- 2 , - 7\right)$

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