Question

The vertices of a rectangle are $\mathrm{E}(-1,1),\mathrm{F}(-1,3),\mathrm{G}(-5,3)$ and $\mathrm{H}(-5,1)$. Find the coordinates of the figure after reflecting in the $\mathrm{x}-$axis, and then translating $3$ units right.

Answer 1

Step 1: Find the coordinates of image after reflection in the $\mathrm{x}-$axis.

Given, coordinates are $\mathrm{E}(-1,1),\mathrm{F}(-1,3),\mathrm{G}(-5,3)$ and $\mathrm{H}(-5,1)$

As we know that reflection about the $\mathrm{x}-$axis then coordinate changes as:

$\mathrm{A}(\mathrm{x},\mathrm{y})=\mathrm{A}\text{'}(\mathrm{x},-\mathrm{y})$

Using the criteria of reflection about $\mathrm{x}-$axis:

$$\begin{gathered} \mathrm{E}(-1,1)=\mathrm{E}\text{'}(-1,-1) \\ \mathrm{F}(-1,3)=\mathrm{F}\text{'}(-1,-3) \\ \mathrm{G}(-5,3)=\mathrm{G}\text{'}(-5,-3) \\ \mathrm{H}(-5,1)=\mathrm{H}\text{'}(-5,-1) \end{gathered}$$

Step 2: Find the coordinates of image after translation.

Coordinates obtained in the above step are $\mathrm{E}\text{'}(-1,-1),\mathrm{F}\text{'}(-1,-3),\mathrm{G}\text{'}(-5,-3),\mathrm{H}\text{'}(-5,-1)$ and $\mathrm{a}=3,\mathrm{b}=0$

Using the above criteria of translation are,

$$\begin{gathered} \mathrm{E}\text{'}(-1,-1)=\mathrm{E}\text{'}\text{'}(-1+3,-1+0)=\mathrm{E}\text{'}\text{'}(2,-1) \\ \mathrm{F}\text{'}(-1,-3)=\mathrm{F}\text{'}\text{'}(-1+3,-3+0)=\mathrm{F}\text{'}\text{'}(2,-3) \\ \mathrm{G}\text{'}(-5,-3)=\mathrm{G}\text{'}\text{'}(-5+3,-3+0)=\mathrm{G}\text{'}\text{'}(-2,-3) \\ \mathrm{H}\text{'}(-5,-1)=\mathrm{H}\text{'}\text{'}(-5+3,-1+0)=\mathrm{H}\text{'}\text{'}(-2,-1) \end{gathered}$$

Step 3: Final answer.

Hence, the coordinates of image after reflection in the $\mathrm{x}-$axis are $\mathrm{E}\text{'}(-1,-1),\mathrm{F}\text{'}(-1,-3),\mathrm{G}\text{'}(-5,-3),\mathrm{H}\text{'}(-5,-1)$ and after translation the coordinates are $\mathrm{E}\text{'}\text{'}(2,-1),\mathrm{F}\text{'}\text{'}(2,-3),\mathrm{G}\text{'}\text{'}(-2,-3),\mathrm{H}\text{'}\text{'}(-2,-1)$.