Formula for Point Reflection over Origin

A point reflection is just a type of reflection. In standard reflections, we reflect over a line, like the y-axis or the x-axis. For a point reflection, we actually reflect over a specific point, usually that point is the origin .

$ \text{Formula} \\ r_{(origin)} \\ (a,b) \rightarrow ( \red -a , \red -b) $

$ r_{origin} (1,2) = (\red -1 , \red -2) $

graph of point reflected in origin

$ r_{origin} (3,4) = (\red -3 , \red -4) $

 example 2 graph of point reflected in origin

Eample 2 shows the same reflection over origin .

Distance to point of reflection

The distance from the preimage to the point of reflection is equal to the distance from the point of reflection to the image .

 point of reflection is the midpoint

A graph showing a triangle’s shape reflected over the origin. The distance from each vertex (and actually each and every single point ) of the preimage to the origin is equal to distance between the origin and each point of the image .

triangle over origin

The origin might be the most common point of reflection, but you can use any point. And the same rules apply. The diagram below uses the point $$(1,2)$$ as the point of reflection.

The the distances between each point on the preimage and the point of reflection $$ (1,2)$$ are equal to the distances between $$(1,2)$$ and each point on the imagetriangle over the point 1,2

Practice problems like example 5 here.

Interactive applet for Point Reflection over Point

Select Shape To reflect

You can drag point of reflection

You can drag around the point to reflect

Practice Problems I

Problem 1

Reflect the point $$ (2,3)$$ over the origin.

reflection problem

Problem 2

Reflect the point $$ (-1,2)$$ over the origin.

$ r_{origin} (-1, 2) \\ (-1, 2) \rightarrow (\red – -1, \red- 2 ) \\ \boxed{ (1,-2 ) } $

reflection problem

Problem 3

Reflect the point $$ (3, -4 )$$ over the origin.

$ r_{origin} (3, -4) \\ (3, -4) \rightarrow (\red -3, \red- -4) \\ \boxed{ (-3,4) } $

Practice Problems II

These practice problems involve reflections over a point that is not the origin like example 5 above.

Problem 2.1

Reflect the point $$ (3,4 )$$ over the point $$ \red { ( 1, 3 )} $$.

There is no simple formula for a reflection over a point like this, but we can follow the 3 steps below to solve this type of question.

First, plot the point of reflection, as shown below.

reflection problem

Problem 2.2

Reflect the point $$ (-5,3 )$$ over the point $$ \red { ( -2, 4 )} $$.

There is no simple formula for a reflection over a point like this, but we can follow the 3 steps below to solve this type of question.

First, plot the point of reflection, as shown below.

reflection problem

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