A line segment has endpoints at (9 ,6 ) and (5 ,3). If the line segment is rotated about the origin by pi , translated vertically by 2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jul 30, 2016

$\left(9 , 6\right) \to \left(- 9 , 4\right) , \left(5 , 3\right) \to \left(- 5 , 1\right)$

Explanation:

Since there are 3 transformations to be performed, name the endpoints A(9 ,6) and B(5 ,3). This will enable us to 'track' the position of the points after each transformation.

First transformation Under a rotation about the origin of $\pi$

a point (x ,y) → (-x ,-y)

hence A(9 ,6) → A'(-9 ,-6) and B(5 ,3) → B'(-5 ,-3)

Second transformation Under a translation $\left(\begin{matrix}0 \\ 2\end{matrix}\right)$

a point (x ,y) → (x ,y+2)

hence A'(-9 ,-6) → A''(-9 ,-4) and B'(-5 ,-3) → B''(-5 ,-1)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(-9 ,-4) → A'''(-9 ,4) and B''(-5 ,-1) → B'''(-5 ,1)

Thus after the 3 transformations.

$\left(9 , 6\right) \to \left(- 9 , 4\right) \text{ and } \left(5 , 3\right) \to \left(- 5 , 1\right)$