Projection and Reflection in Coordinate Geometry
Lesson · Advanced
Geometry: Coordinate Geometry
Let
and let the line be
Projection onto a Line
Let H be the perpendicular projection of P onto the line.
Then
Reflection Across a Line
If P'=(x',y') is the reflection of P across
then
A normal vector to the line is
so the vector from P to its projection H must be parallel to (a,b). Thus, for some t,
Since H lies on the line,
Therefore,
so
Hence
Since H is the midpoint of P and its reflection P',
which immediately gives
Let
and
Let H be the projection of P onto ℓ, and let P' be the reflection of P across ℓ. Find both H and P'.
We have
and
Thus
Since H is the midpoint of PP',
so
Therefore,
Point A=(2,7) is reflected across a line to A'=(8,-1). Find the equation of the line.
The reflection line is the perpendicular bisector of AA'.
The midpoint is
The slope of AA' is
so the reflection line has slope 3/4. Therefore,
or
Reflect the line
across the line y=x. Find the equation of its image.
Reflection across y=x exchanges the coordinates:
Therefore, interchange x and y in the original equation:
Hence the reflected line is
Reflect the line
across the line
Find the equation of the reflected line.
For reflection across
we have a=1, b=2. Thus
so
Since reflection is its own inverse,
Substitute these into
to get
which simplifies to
Therefore, the reflected line is