Question

The vertices of a triangle are $\mathrm{A}(2,5),\mathrm{B}(1,2),\mathrm{C}(3,1)$. Find the coordinates of the image after the transformations given.

Reflect in the $\mathrm{x}-$axis and then rotate $90°$ counterclockwise about the origin.

Answer 1

Step 1: Find the coordinates of image after reflection

Given, coordinates are $\mathrm{A}(2,5),\mathrm{B}(1,2),\mathrm{C}(3,1)$.

As we know reflection in the $\mathrm{x}-$axis then the coordinates changes as:

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

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

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

Step 2: Find the coordinates of image after rotation.

As we have obtained coordinates in the above step are, $\mathrm{A}\text{'}(2,-5),\mathrm{B}\text{'}(1,-2),\mathrm{C}\text{'}(3,-1)$

Since, we know for rotation of $90°$ counterclockwise about the origin then the coordinate changes as,

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

Using the above criteria of rotation of $90°$ counterclockwise about the origin,

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

Step 3: Final answer.

Hence, the coordinates of final image are $\mathrm{A}\text{'}\text{'}(2,5),\mathrm{B}\text{'}\text{'}(1,2),\mathrm{C}\text{'}\text{'}(3,1)$.