A line segment has endpoints at (3 , 2) and (1 ,3). If the line segment is rotated about the origin by (3pi)/2 , translated horizontally by -5, and reflected about the x-axis, what will the line segment's new endpoints be?

Dec 30, 2017

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

Explanation:

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

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

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

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

• " a point "(x,y)to(y,-x)

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

$\Rightarrow B \left(1 , 3\right) \to B ' \left(3 , - 1\right)$

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

$\text{under a translation } \left(\begin{matrix}- 5 \\ 0\end{matrix}\right)$

• " a point "(x,y)to(x-5,y)

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

$\Rightarrow B ' \left(3 , - 1\right) \to B ' ' \left(- 2 , - 1\right)$

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

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

• " a point "(x,y)to(x,-y)

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

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

$\text{under all 3 transformations}$

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