Hands-on math!

Cartesian Coordinates - Reflections about a Line

You are often asked to reflect (or rotate) a point or a polygon about a line. Usually this is either the x-axis, the y-axis or the lines y = x or y = -x. Here are the rules for those reflections:




Reflection
Rule		Example
(see figure)→
About the y-axis Negate the x coordinate:
(x,y) → (-x,y)		 Point A → B
(-5,-2) → (5,-2)
About the x-axis Negate the y coordinate:
(x,y) → (x,-y)		 Point B → C
(5,-2) → (5,2)
About the line
y = x		 Exchange coordinates:
(x,y) → (y,x)		 Point C → D
(5,2) → (2,5)
About the line
y = -x		 Exchange and
negate both coordinates:
(x,y) → (-y,-x)		 Point E → F
(-2,5) → (-5,2)

Suppose you wanted to reflect around a line that is parallel to one of the above 4 defined lines: x-axis, y-axis, y = x or y = -x?

Here's how you do that:
  • Compute the translation in either x or y that makes the reflection line cross the origin:
          either:
    • (dx,0)   <== moves the line left or right dx spaces or
    • (0,dy)   <== moves the line up or down dy spaces.
  • Translate the point(s) by this same translation
  • Reflect the translated point(s) as shown in the table above
  • Translate back to the original position by either:
    • (-dx,0)   <== if you translated the line in x or
    • (0,-dy)   <== if you translated the line in y
Here's an example:
Reflect (2,2) around the line y = x + 5,
  • Compute the translation: The translation you would use is (0,-5)
    (this makes the line y = x which goes through the origin)
  • Translate the point:
        (2,2) + (0,-5) = (2,-3)
  • Reflect the translated point:
    (2,-3) becomes (-3,2)
    (See the above table).
  • Translate back:
    The opposite of (0,-5) is (0,+5):
    (-3,2) + (0,+5) = (-3,7) Get it?

Now, here are some of problems for you:
1). Reflect point A about y = x   (-5,-2) → (___,___)  Plot this point on the grid.
2). Reflect point B about y = - x   (5,-2) → (___,___)  Plot this point on the grid.
3). Reflect point D about y = - x   (2,5) → (___,___)  Plot this point on the grid.
What did point D become under the reflection about y = - x? ___________

Note: It is possible to reflect points about other lines, even arbitrary lines like y = 2x, but that requires trigonometry, a subject you will learn in high school!