Question

A point $P$ is mapped onto $P^{\prime }$ under the reflection in the
$x$-axis. $P^{\prime }$ is mapped onto $P^{\prime \prime }$ under the reflection in the origin. If the co-ordinates of $P^{\prime \prime }$ are $(5,-2)$, write down the co-ordinates of $P$. State the single transformation that takes place.

Answer 1

Step $1$: Find the coordinates of $P^{\prime }$

Given: Coordinates of $P^{\prime \prime }=(5,-2)$

$P^{\prime }$ is mapped onto $P^{\prime \prime }$ under the reflection in the origin.

Reflection in the origin, $M_o(x,y)=(-x,-y)$

$\Rightarrow M_o(x,y)=(5,-2)$

So,

$-x=5\Rightarrow x=-5$

$-y=-2\Rightarrow y=2$

$\therefore$The co-ordinates of $P^{\prime }$ are $(-5,2)$.

Step $2$: Find the coordinates of $P$

$P$ is mapped onto $P^{\prime }$ under the reflection in the
$x$-axis.

Reflection in the $x$-axis, $M_x(x,y)=(x,-y)$

$\Rightarrow M_x(x,y)=(-5,2)$

So,

$x=-5$

$-y=2\Rightarrow y=-2$

$\therefore$The co-ordinates of $P$ are $(-5,-2)$.

Step $3$: State the single transformation that takes place

Co-ordinates of $P=(-5,-2)$

Co-ordinates of $P^{\prime \prime }=(5,-2)$

On reflection, only the sign of the abscissa
($x$-coordinate) has changed.

Reflection in the $y$-axis, $M_y(x,y)=(-x,y)$

Hence, the single transformation that maps $P$ onto $P^{\prime \prime }$ is reflection in the $y$-axis.