Question

The point A(4, 1) is reflected over the point (0, -3) and its image is point B. What are the coordinates of point B?

Answer 1

A reflection over the point (0,0) is given by:

$(x,y)\rightarrow(-x,-y)$

Nevertheless, the reflection is being made over the point (0,-3). We can first make a traslation of the point (0,-3) to the origin. Then, reflect the new coordinates of the point A through the origin and then bring back the new origin to (0,-3).

First, make a traslation such that the new coordinates of (0,-3) are (0,0):

$(x,y)\rightarrow(x,y+3)$

Then, apply a reflection through the origin:

$(x,y+3)\rightarrow(-x,-y-3)$

Finally, make a traslation reciprocal to the first one:

$(-x,-y-3)\rightarrow(-x,-y-3-3)=(-x,-y-6)$

Therefore, the reflection over the point (0,-3) can be written as:

$(x,y)\rightarrow(-x,-y-6)$

Substitute (x,y)=(4,1) to find out the new coordinates of A after the reflection:

$\begin{gathered} A\rightarrow B \\ \Rightarrow(4,1)\rightarrow(-4,-1-6)=(-4,-7) \end{gathered}$

Therefore, the coordinates of the point B are (-4,-7).