# A line segment has endpoints at (5 ,8 ) and (7 ,4). If the line segment is rotated about the origin by pi , translated horizontally by -1 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jan 21, 2016

$\left(6 , - 8\right) , \left(8 , - 4\right)$

#### Explanation:

We can create a rule for this transformation.

Assume the original endpoints of the segment can be described as color(red)((x,y).

The first way in which the segment is manipulated is with a rotation of $\pi$, or ${180}^{\circ}$. This translates by taking the opposite of the $x$ and $y$ values of the point, or our new "rule" for this point in the transformation: color(red)((-x,-y)

The endpoints are then translated horizontally by $- 1$. Horizontal movement corresponds to the $x$ coordinate, whereas vertical corresponds to the $y$ coordinate. Since this has been horizontally shifted, the new rule, working off the most previous rule, is: color(red)((-x-1,-y)

The final transformation is a reflection over the $y$ axis, which means that the $x$ coordinate's sign is flipped: color(red)((x+1,-y)

This is the rule for the entire transformation. Apply it to the points $\left(5 , 8\right)$ and $\left(7 , 4\right)$.

$\left(5 , 8\right) \rightarrow \left(6 , - 8\right)$

$\left(7 , 4\right) \rightarrow \left(8 , - 4\right)$