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

Jan 28, 2018

The new endpoints are $\left(1 , 1\right)$ and $\left(2 , 3\right)$.

Explanation:

Let's apply each of the transformations to both points, one at a time:

$\pi$ rotation around the origin (180º):
$\left(x , y\right) \implies \left(- x , - y\right)$
$\left(3 , 1\right) \implies \left(- 3 , - 1\right)$
$\left(2 , 3\right) \implies \left(- 2 , - 3\right)$

Horizontal translation by $4$:
$\left(x , y\right) \implies \left(x + 4 , y\right)$
$\left(- 3 , - 1\right) \implies \left(1 , - 1\right)$
$\left(- 2 , - 3\right) \implies \left(2 , - 3\right)$

Reflection over the $x$-axis:
$\left(x , y\right) \implies \left(x , - y\right)$
$\left(1 , - 1\right) \implies \left(1 , 1\right)$
$\left(2 , - 3\right) \implies \left(2 , 3\right)$

The new endpoints are $\left(1 , 1\right)$ and $\left(2 , 3\right)$.

Jan 28, 2018

(1,1)" and "((2,3)

Explanation:

$\text{since there are 3 transformations to be performed here}$

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

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

$\text{under a rotation about the origin by } \pi$

• " a point "(x,y)to(-x,-y)

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

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

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

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

• " a point "(x,y)to(x+4,y)

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

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

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

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

• " a point "(x,y)to(x,-y)

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

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

$\text{after all 3 transformations}$

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