Question

Points $(-5,0)$ and $(4,0)$ are invariant points under reflection in the line $L_1$; points $(0,-6)$ and $(0,5)$ are invariant on reflection in the line $L_2$.
(a) Name or write equations for the lines $L_1$ and $L_2$.
(b) Write down the images of $P(2,6)$ and $Q(-8,-3)$ on reflection in $L_1$. Name the images as $P^{\prime }$ and $Q^{\prime }$ respectively.
(c) Write down the images of $P$ and $Q$ on reflection in $L_2$. Name the images as $P^{\prime \prime }$ and $Q^{\prime \prime }$ respectively.
(d) State or describe a single transformation that maps $Q^{\prime }$ onto $Q^{\prime \prime }$.

Answer 1

(a) We know that every point in a line is invariant under the reflection in the same line.
Since, points $(-5,0)$ and $(4,0)$ lie on the $x$-axis.
$\Rightarrow$ Points $(-5,0)$ and $(4,0)$ are invariant under reflection in $x$-axis.
Given that the points $(-5,0)$ and $(4,0)$ are invariant on reflection in line $L_1$.
$\therefore$ The line $L_1$ is $x$-axis, whose equation is $y=0$
Similarly, the given points $(0,-6)$ and $(0,5)$ lie on the $y$-axis and are invariant on reflection in line $L_2$.
$\therefore$ The line $L_2$ is $y$-axis, whose equation is $x=0$
(b) $P^{\prime }=$ The image of $P(2,6)$ in $L_1$
$=$ The image of $P(2,6)$ in $x$-axis $=(2,-6)$
And, $Q^{\prime }=$ The image of $Q(-8,-3)$ in $L_1$
$=$ The image of $Q(-8,-3)$ in $x$-axis $=(-8,3)$
(c) $P^{\prime \prime }=$ The image of $P(2,6)$ in $L_2$
$=$ The image of $P(2,6)$ in $y$-axis $=(-2,6)$
$Q^{\prime \prime }=$ The image of $Q(-8,-3)$ in $L_2$
$=$ The image of $Q(-8,-3)$ in $y$-axis $=(8,-3)$
(d) Since, $Q^{\prime }=(-8,3)$ and $Q^{\prime \prime }=(8,-3)$
and we know $M_0(-x,y)=(x,-y)$
$\therefore$ The single transformation that maps $Q^{\prime }$ onto $Q^{\prime \prime }=$ Reflection in origin.