Question

Write down the co-ordinates of the image of the point $(-2,4)$ under reflection in the origin and under reflection in the $y$-axis. What is the distance between the points of reflection?

Answer 1

Step $1$: Find the co-ordinates of the images

Let the point be $R$.

Given: Co-ordinates of $R=(-2,4)$

Let $R$ is reflected in the origin as $R^{\prime }$.

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

$\Rightarrow M_o(-2,4)=(2,-4)$

$\Rightarrow R^{\prime }=(2,-4)$

Let $R$ is reflected in the $y$-axis as $R^{\prime \prime }$.

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

$\Rightarrow M_y(-2,4)=(2,4)$

$\Rightarrow R^{\prime \prime }=(2,4)$

Step $2$: Find the distance between the points of reflection

Co-ordinates of $R^{\prime }=(2,-4)$

Co-ordinates of $R^{\prime \prime }=(2,4)$

The distance between points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given

by $AB=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.

$\therefore R^{\prime }{R}^{\prime \prime }=\sqrt{(2-2{)}^2+(4-(-4){)}^2}$

$=\sqrt{(0{)}^2+(8{)}^2}$

$=\sqrt{64}$

$=8\text{units}$