Question

The vertices of a triangle are $\mathrm{A}(2,5),\mathrm{B}(1,2)$ and $\mathrm{C}(3,1)$. Find the coordinates of the image after the transformations given. Dilate with respect to the origin using a scale factor of $2$. Then, translation $2$ units left and $1$ unit up.

Answer 1

Step 1:Find the coordinates of image after dilation.

Given, coordinates are $\mathrm{A}(2,5),\mathrm{B}(1,2)$ and $\mathrm{C}(3,1)$ with scale factor of $2$.

Since, we know dilation of scale $k$ then the coordinate changes as:

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

Using the above criteria of dilating the figure by scale factor of $2$.

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

Step 2: Find the coordinates of image after translation.

Coordinates obtained in above step are $\mathrm{A}\text{'}(4,10),\mathrm{B}\text{'}(2,4),\mathrm{C}\text{'}(6,2)$.

Now, using the criteria of translation by parameters, $\mathrm{a}=-2,\mathrm{b}=1$

.$$\begin{gathered} \mathrm{A}\text{'}(4,10)=\mathrm{A}\text{'}\text{'}(4-2,10+1)=\mathrm{A}\text{'}\text{'}(2,11) \\ \mathrm{B}\text{'}(2,4)=\mathrm{B}\text{'}\text{'}(2-2,4+1)=\mathrm{B}\text{'}\text{'}(0,5) \\ \mathrm{C}\text{'}(6,2)=\mathrm{C}\text{'}\text{'}(6-2,2+1)=\mathrm{C}\text{'}\text{'}(4,3) \end{gathered}$$

Step 3: Final answer.

Hence, the coordinates of final image are $\mathrm{A}\text{'}\text{'}(2,11),\mathrm{B}\text{'}\text{'}(0,5),\mathrm{C}\text{'}\text{'}(4,3)$