Question

A point $P$ is reflected in the $x$-axis to $P^{\prime }.P^{\prime }$ is then reflected in the origin to $P^{\prime \prime }$. If the co-ordinates of $P^{\prime }$ are $(-3,4)$. Find the co-ordinates of $P$ and $P^{\prime \prime }$. Write the single transformation that map $P$ onto $P^{\prime \prime }$.

Answer 1

Step $1$: Find the coordinates of $P$

Given: Coordinates of $P^{\prime }=(-3,4)$

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

$P^{\prime }$ is the image of $P$ reflected in the $x$-axis.

$\Rightarrow M_x(x,y)=(-3,4)$

So,

$x=-3$

$-y=4\Rightarrow y=-4$

Hence, the co-ordinates of P are $(-3,-4)$.

Step $2$: Find the coordinates of $P^{\prime \prime }$

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

$P^{\prime \prime }$ is the image of $P^{\prime }$ reflected in the origin.

$\Rightarrow M_o(-3,4)=(3,-4)$

Hence, the co-ordinates of $P^{\prime \prime }$ are $(3,-4)$.

Step $3$: Write the single transformation that map $P$ onto $P^{\prime \prime }$

Co-ordinates of $P=(-3,-4)$

Co-ordinates of $P^{\prime \prime }=(3,-4)$

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 the reflection in the $y$-axis.