A line segment has endpoints at (2 ,5 ) and (7 , 3 ). If the line segment is rotated about the origin by  pi /2 , translated horizontally by  1 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jul 12, 2017

4,2)" and " (2,7)

Explanation:

Since there are 3 transformations to be performed label the endpoints A(2,5) and B(7,3)

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(2 , 5\right) \to A ' \left(- 5 , 2\right) , B \left(7 , 3\right) \to B ' \left(- 3 , 7\right)$

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

$\text{under a translation } \left(\begin{matrix}1 \\ 0\end{matrix}\right)$

• " a point "(x,y)to(x+1,y)

$\Rightarrow A ' \left(- 5 , 2\right) \to A ' ' \left(- 4 , 2\right)$

$\Rightarrow B ' \left(- 3 , 7\right) \to B ' ' \left(- 2 , 7\right)$

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

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

• " a point "(x,y)to(-x,y)

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

$\Rightarrow B ' ' \left(- 2 , 7\right) \to B ' ' ' \left(2 , 7\right)$

$\text{hence after all 3 transformations}$

$\left(2 , 5\right) \to \left(4 , 2\right) \text{ and } \left(7 , 3\right) \to \left(2 , 7\right)$