Question

Translate a point $(\mathrm{x},\mathrm{y})$ $3$ units left and $5$ units up. Then translate the image $5$ units right and $2$ units up. What are the coordinates of the point after the translations?

Answer 1

Find the coordinates of image after translation.

As we have given, $\mathrm{A}(\mathrm{x},\mathrm{y})$ and $\mathrm{a}=-3,\mathrm{b}=5$.

Since, we know that to translate a figure $\text{'}\mathrm{a}\text{'}$ units horizontally and $\text{'}\mathrm{b}\text{'}$ units vertically in the coordinate plane. $\text{'}\mathrm{a}\text{'}$ is added to $\mathrm{x}-$coordinate and $\text{'}\mathrm{b}\text{'}$ is added to the $\mathrm{y}-$coordinate of the vertices.

$\mathrm{A}(\mathrm{x},\mathrm{y})=\mathrm{A}\text{'}(\mathrm{x}+\mathrm{a},\mathrm{y}+\mathrm{b})$

The value of $\text{'}\mathrm{a}\text{'}$ and $\text{'}\mathrm{b}\text{'}$ will be positive if the shift is right and vertical up respectively and the value of $\text{'}\mathrm{a}\text{'}$ and $\text{'}\mathrm{b}\text{'}$ will be negative if the shift is left and vertical down respectively.

Using the above criteria for translation,

$\mathrm{A}\text{'}(\mathrm{x}+\mathrm{a},\mathrm{y}+\mathrm{b})=\mathrm{A}\text{'}(\mathrm{x}-3,\mathrm{y}+5)$

Now. again translating image $5$ unit's right and $2$ unit's up.

Final image after first translation for $\mathrm{A}\text{'}(\mathrm{x}-3,\mathrm{y}+5)$ and $\mathrm{a}=5,\mathrm{b}=2$

$\mathrm{A}\text{'}(\mathrm{x}-3,\mathrm{y}+5)=\mathrm{A}\text{'}\text{'}(\mathrm{x}-3+5,\mathrm{y}+5+5)=\mathrm{A}\text{'}\text{'}(\mathrm{x}+2,\mathrm{y}+7)$

Hence, the final image is $\mathrm{A}\text{'}\text{'}(\mathrm{x}+2,\mathrm{y}+7)$.