# A line segment has endpoints at (3 , 4) and (2 , 6). If the line segment is rotated about the origin by (3pi)/2 , translated horizontally by -2, and reflected about the y-axis, what will the line segment's new endpoints be?

May 19, 2018

$\left(- 2 , - 3\right) \text{ and } \left(- 4 , - 2\right)$

#### Explanation:

$\text{since there are 3 transformations to be performed label}$
$\text{the endpoints}$

$A = \left(3 , 4\right) \text{ and } B = \left(2 , 6\right)$

$\textcolor{b l u e}{\text{First transformation}}$

$\text{under a rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(3 , 4\right) \to A ' \left(4 , - 3\right)$

$\Rightarrow B \left(2 , 6\right) \to B ' \left(6 , - 2\right)$

$\textcolor{b l u e}{\text{Second transformation}}$

$\text{under a horizontal translation } \left(\begin{matrix}- 2 \\ 0\end{matrix}\right)$

• " a point "(x,y)to(x-2,y)

$\Rightarrow A ' \left(4 , - 3\right) \to A ' ' \left(2 , - 3\right)$

$\Rightarrow B ' \left(6 , - 2\right) \to B ' ' \left(4 , - 2\right)$

$\textcolor{b l u e}{\text{Third transformation}}$

$\text{under a reflection in the y-axis}$

• " a point "(x,y)to(-x,y)

$\Rightarrow A ' ' \left(2 , - 3\right) \to A ' ' ' \left(- 2 , - 3\right)$

$\Rightarrow B ' ' \left(4 , - 2\right) \to B ' ' ' \left(- 4 , - 2\right)$

$\text{After all 3 transformations}$

3,4)to(-2,-3)" and "(2,6)to(-4,-2)