# A line segment has endpoints at (8 , 4) and (1 , 2). If the line segment is rotated about the origin by (pi)/2 , translated vertically by 4, and reflected about the x-axis, what will the line segment's new endpoints be?

May 22, 2018

$\left(4 , - 12\right)$ and $\left(2 , - 5\right)$

#### Explanation:

You have not specified the direction of the rotation, so I will take this as being counter clockwise. If we look at each transformation and translation in order we get the following:

A rotation of $\frac{\pi}{2}$ anticlockwise will map:

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

This result takes a little thought. The easiest way to see this is, to think not of rotating a point, but rotating the axes themselves. If we rotate the axes $\frac{\pi}{2}$ counterclockwise, the positive x axis becomes the positive y axis and the positive y axis becomes the negative x axis.

A translation of 4 units vertically maps:

$\left(x , y\right) \to \left(x , y + 4\right)$

A reflection in the x axis maps:

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

This is the same as reflecting the axes, so positive y becomes negative y and x remains unchanged.

Putting these together in order:

$\left(x , y\right) \to \left(- y , x\right) \to \left(- y , x + 4\right) \to \left(y , - \left(x + 4\right)\right)$

Naming endpoints A and B:

$A = \left(8 , 4\right)$

$B = \left(1 , 2\right)$

$A = \left(8 , 4\right) \to \left(y , - \left(x + 4\right)\right) = \left(4 , - 12\right)$

$B = \left(1 , 2\right) \to \left(y , - \left(x + 4\right)\right) = \left(2 , - 5\right)$