A line segment has endpoints at (5 , 8) and (6 , 1). If the line segment is rotated about the origin by pi/2 , translated vertically by -2, and reflected about the y-axis, what will the line segment's new endpoints be?

May 21, 2018

Rule for Rotation by $\frac{\pi}{2}$ or 90 degrees Counter-Clockwise:
$\left(x , y\right) \to \left(- y , x\right)$

Rule for Translated Vertically by $- 2$
$\left(x , y\right) \to \left(x , y - 2\right)$

Rule for Reflection across y-axis
$\left(x , y\right) \to \left(- x , y\right)$

In that same order:
$\left(5 , 8\right) \to \left(- 8 , 5\right) \to \left(- 8 , 3\right) \to \left(8 , 3\right)$
$\left(6 , 1\right) \to \left(- 1 , 6\right) \to \left(- 1 , 4\right) \to \left(1 , 4\right)$

To Check, each of these transformations are isometric which means the distance between the points will not change.

Original Points
$\sqrt{{\left(5 - 6\right)}^{2} + {\left(8 - 1\right)}^{2}} = \sqrt{{\left(- 1\right)}^{2} + {\left(7\right)}^{2}} = \sqrt{1 + 49} = \sqrt{50}$

New Points
$\sqrt{{\left(8 - 1\right)}^{2} + {\left(3 - 4\right)}^{2}} = \sqrt{{\left(7\right)}^{2} + {\left(- 1\right)}^{2}} = \sqrt{49 + 1} = \sqrt{50}$

They have the same distance so the segment endpoints are correct.

May 21, 2018

$\left(8 , 3\right) \text{ and } \left(1 , 4\right)$

Explanation:

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

$A = \left(5 , 8\right) \text{ and } B = \left(6 , 1\right)$

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(5 , 8\right) \to A ' \left(- 8 , 5\right)$

$\Rightarrow B \left(6 , 1\right) \to B ' \left(- 1 , 6\right)$

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

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

• " a point "(x,y)to(x,y-2)

$\Rightarrow A ' \left(- 8 , 5\right) \to A ' ' \left(- 8 , 3\right)$

$\Rightarrow B ' \left(- 1 , 6\right) \to B ' ' \left(- 1 , 4\right)$

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

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

• " a point "(x,y)to(-x,y)

$\Rightarrow A ' ' \left(- 8 , 3\right) \to A ' ' ' \left(8 , 3\right)$

$\Rightarrow B ' ' \left(- 1 , 4\right) \to B ' ' ' \left(1 , 4\right)$

$\text{After all 3 transformations}$

$\left(5 , 8\right) \to \left(8 , 3\right) \text{ and } \left(6 , 1\right) \to \left(1 , 4\right)$