Question

The vertices of a figure are given. Find the coordinates of the image after the transformations given.

$\mathrm{A}(-5,3),\mathrm{B}(-2,3),\mathrm{C}(-2,1),\mathrm{D}(-5,1)$

Reflect in the $\mathrm{y}-$axis. Then, dilate using a scale factor of $2$.

Answer 1

Step 1: Find the coordinates of image after reflection.

Given, $\mathrm{A}(-5,3),\mathrm{B}(-2,3),\mathrm{C}(-2,1),\mathrm{D}(-5,1)$

As we know that reflection about $\mathrm{y}-$axis, then coordinates changes as,

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

Using the above criteria of reflection about $\mathrm{y}-$axis.

$$\begin{gathered} \mathrm{A}(-5,3)=\mathrm{A}\text{'}(5,3) \\ \mathrm{B}(-2,3)=\mathrm{B}\text{'}(2,3) \\ \mathrm{C}(-2,1)=\mathrm{C}\text{'}(2,1) \\ \mathrm{D}(-5,1)=\mathrm{D}\text{'}(5,1) \end{gathered}$$

Step 2: Find the coordinates of image after dilation.

From the above step, coordinate of the image are $\mathrm{A}\text{'}(5,3),\mathrm{B}\text{'}(2,3),\mathrm{C}\text{'}(2,1),\mathrm{D}\text{'}(5,1)$

Since, we know that in dilation we just multiply each $\mathrm{x}$ and $\mathrm{y}$ coordinate by the given scale factor.

$\mathrm{P}(\mathrm{x},\mathrm{y})=\mathrm{P}\text{'}(\mathrm{x}\times \mathrm{a},\mathrm{y}\times \mathrm{a})$

Using the above criteria of dilation using a scale factor of $2$.

$$\begin{gathered} \mathrm{A}\text{'}(5,3)=\mathrm{A}\text{'}\text{'}(5\times 2,3\times 2)=\mathrm{A}\text{'}\text{'}(10,6) \\ \mathrm{B}\text{'}(2,3)=\mathrm{B}\text{'}\text{'}(2\times 2,3\times 2)=\mathrm{B}\text{'}\text{'}(4,6) \\ \mathrm{C}\text{'}(2,1)=\mathrm{C}\text{'}\text{'}(2\times 2,1\times 2)=\mathrm{C}\text{'}\text{'}(4,2) \\ \mathrm{D}\text{'}(5,1)=\mathrm{D}\text{'}\text{'}(5\times 2,1\times 2)=\mathrm{D}\text{'}\text{'}(10,2) \end{gathered}$$

Step 3: Final answer.

Hence, the coordinates of image after reflection are $\mathrm{A}\text{'}(5,3),\mathrm{B}\text{'}(2,3),\mathrm{C}\text{'}(2,1),\mathrm{D}\text{'}(5,1)$ and after dilation are $\mathrm{A}\text{'}\text{'}(10,6),\mathrm{B}\text{'}\text{'}(4,6),\mathrm{C}\text{'}\text{'}(4,2),\mathrm{D}\text{'}\text{'}(10,2)$.