# A line segment has endpoints at (7 ,4 ) and (3 ,5 ). If the line segment is rotated about the origin by (3 pi)/2 , translated vertically by -2 , and reflected about the y-axis, what will the line segment's new endpoints be?

Aug 24, 2016

$\left(7 , 4\right) \to \left(- 4 , - 9\right) \text{ and } \left(3 , 5\right) \to \left(- 5 , - 5\right)$

#### Explanation:

Since there are 3 transformations to be performed, name the endpoints A(7 ,4) and B(3 ,5) so that we can 'track' the coordinates of the endpoints after each transformation.

First transformation Under a rotation about the origin of $\frac{3 \pi}{2}$

a point (x ,y) → (y ,-x)

hence A(7 ,4) → A'(4 ,-7) and B(3 ,5) → B'(5 ,-3)

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

a point (x ,y) → (x ,y-2)

hence A'(4 ,-7) → A''(4 ,-9) and B'(5 ,-3) → B''(5 ,-5)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

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

Thus after all 3 transformations the endpoints are.

$\left(7 , 4\right) \to \left(- 4 , - 9\right) \text{ and } \left(3 , 5\right) \to \left(- 5 , - 5\right)$