A line segment has endpoints at (5 ,8 ) and (7 ,6). If the line segment is rotated about the origin by pi , translated horizontally by -3 , and reflected about the y-axis, what will the line segment's new endpoints be?

Aug 15, 2016

$\left(5 , 8\right) \to \left(8 , - 8\right) \text{ and } \left(7 , 6\right) \to \left(10 , - 6\right)$

Explanation:

Since there are 3 transformations to be performed here, name the endpoints A(5 ,8) and B(7 ,6) so we can 'track' the coordinates after each transformation.

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

a point (x ,y) → (-x ,-y)

hence A(5 ,8) → A'(-5 ,-8) and B(7 ,6) → B'(-7 ,-6)

Second transformation Under a translation of $\left(\begin{matrix}- 3 \\ 0\end{matrix}\right)$

a point (x ,y) → (x-3 ,y)

hence A'(-5 ,-8) → A''(-8 ,-8) and B'(-7 ,-6) → B''(-10 ,-6)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-8 ,-8) → A'''(8 ,-8) and B''(-10 ,-6) → B'''(10 ,-6)

Thus after all 3 transformations.

$\left(5 , 8\right) \to \left(8 , - 8\right) \text{ and } \left(7 , 6\right) \to \left(10 , - 6\right)$