Question

The vertices of a rectangle are $\mathrm{Q}(-6,2),\mathrm{R}(6,2),\mathrm{S}(6,-4)$ and $\mathrm{T}(-6,-4)$. Dilate the rectangle with respect to the origin using a scale factor of $\frac{3}{2}$.Then translate it $5$ unit's right and $1$ unit down. What are the coordinate of image?

Answer 1

Step 1: Find the coordinates of image after dilation.

Given, coordinates $\mathrm{Q}(-6,2),\mathrm{R}(6,2),\mathrm{S}(6,-4)$ and $\mathrm{T}(-6,-4)$ and scale factor is $\frac{3}{2}$.

Since, in dilation with scale factor $\mathrm{k}$ 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 for dilation,

$$\begin{gathered} \mathrm{Q}(-6,2)=\mathrm{Q}\text{'}(-6\times \frac{3}{2},2\times \frac{3}{2})=\mathrm{Q}\text{'}(-9,3) \\ \mathrm{R}(6,2)=\mathrm{R}\text{'}(6\times \frac{3}{2},2\times \frac{3}{2})=\mathrm{R}\text{'}(9,3) \\ \mathrm{S}(6,-4)=\mathrm{S}\text{'}(6\times \frac{3}{2},-4\times \frac{3}{2})=\mathrm{S}\text{'}(9,-6) \\ \mathrm{T}(-6,-4)=\mathrm{T}\text{'}(-6\times \frac{3}{2},-4\times \frac{3}{2})=\mathrm{T}\text{'}(-9,-6) \end{gathered}$$

Step 2: Find the coordinates of image after translation.

Coordinates obtained from the above step are $\mathrm{Q}\text{'}(-9,3),\mathrm{R}\text{'}(9,3),\mathrm{S}\text{'}(9,-6),\mathrm{T}\text{'}(-9,-6)$.

For translation, parameters are $\mathrm{a}=5,\mathrm{b}=-1$

$$\begin{gathered} \mathrm{Q}\text{'}(-9,3)=\mathrm{Q}\text{'}\text{'}(-9+5,3-1)=\mathrm{Q}\text{'}\text{'}(-4,2) \\ \mathrm{R}\text{'}(9,3)=\mathrm{R}\text{'}\text{'}(9+5,3-1)=\mathrm{R}\text{'}\text{'}(14,2) \\ \mathrm{S}\text{'}(9,-6)=\mathrm{S}\text{'}\text{'}(9+5,-6-1)=\mathrm{S}\text{'}\text{'}(14,-7) \\ \mathrm{T}\text{'}(-9,-6)=\mathrm{T}\text{'}\text{'}(-9+5,-6-1)=\mathrm{T}\text{'}\text{'}(-4,-7) \end{gathered}$$

Step 3: Final answer.

Hence, the coordinates of image after dilation are $\mathrm{Q}\text{'}(-9,3),\mathrm{R}\text{'}(9,3),\mathrm{S}\text{'}(9,-6),\mathrm{T}\text{'}(-9,-6)$ and after translation are $\mathrm{Q}\text{'}\text{'}(-4,2),\mathrm{R}\text{'}\text{'}(14,2),\mathrm{S}\text{'}\text{'}(14,-7),\mathrm{T}\text{'}\text{'}(-4,-7)$