A line segment has endpoints at (2 ,3 ) and (3 ,9 ). If the line segment is rotated about the origin by ( pi)/2 , translated vertically by -8 , and reflected about the x-axis, what will the line segment's new endpoints be?

Dec 1, 2016

$\left(2 , 3\right) \to \left(3 , 6\right) \text{ and } \left(3 , 9\right) \to \left(- 9 , 5\right)$

Explanation:

Since there are 3 transformations to be performed here, label the endpoints A (2 ,3) and B (3 ,9)

First Transformation Under a rotation about the origin of $\frac{\pi}{2}$

$\text{ a point } \left(x , y\right) \to \left(- y , x\right)$

Hence A(2 ,3) → A'(-3 ,2) and B(3 ,9) → B' (-9 ,3)

Second Transformation Under a translation $\left(\begin{matrix}0 \\ - 8\end{matrix}\right)$

$\text{ a point } \left(x , y\right) \to \left(x , y - 8\right)$

Hence A'(-3 ,2) → A''(3 ,-6) and B'(-9 ,3) → B''(-9 ,-5)

Third transformation Under a reflection in the x-axis

$\text{ a point } \left(x , y\right) \to \left(x , - y\right)$

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

Thus after all 3 transformations.

$\left(2 , 3\right) \to \left(3 , 6\right) \text{ and } \left(3 , 9\right) \to \left(- 9 , 5\right)$