A line segment has endpoints at (2 , 3) and (1 , 2). If the line segment is rotated about the origin by (pi)/2 , translated vertically by 4, and reflected about the x-axis, what will the line segment's new endpoints be?

Dec 9, 2017

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

Explanation:

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

$A \left(2 , 3\right) \text{ and } B \left(1 , 2\right)$

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

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

• " a point "(x,y)to(-y,x)

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

$\Rightarrow B \left(1 , 2\right) \to B ' \left(- 2 , 1\right)$

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

$\text{under a translation } \left(\begin{matrix}0 \\ 4\end{matrix}\right)$

• " a point "(x,y)to(x,y+4)

$\Rightarrow A ' \left(- 3 , 2\right) \to A ' ' \left(- 3 , 6\right)$

$\Rightarrow B ' \left(- 2 , 1\right) \to B ' ' \left(- 2 , 5\right)$

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

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

• " a point "(x,y)to(x,-y)

$\Rightarrow A ' ' \left(- 3 , 6\right) \to A ' ' ' \left(- 3 , - 6\right)$

$\Rightarrow B ' ' \left(- 2 , 5\right) \to B ' ' ' \left(- 2 , - 5\right)$

$\text{after all 3 transformations}$

$\left(2 , 3\right) \to \left(- 3 , - 6\right) \text{ and } \left(1 , 2\right) \to \left(- 2 , - 5\right)$