TRANSFORMATIONS ( Line Reflection)
A transformation of the plane is a one-to-one correspondence
between the points in a plane such that each point is associated with
itself or with some other line in the plane.
Line Reflections –
In the figure, ∆ABC  ∆A’B’C’. One triangle will
“fit exactly” on top of the other by folding along
line k, the line of reflection.
Point A corresponds to point A’. Point B
corresponds to B’.
Point C corresponds to itself and is called a fixed
point.
The term image is used to describe the relationship of these points.
The image of A is A’ and A’ is A
A → A’ and A’ → A
The image of B is B’ and B’ is B
B → B’ and B’ → B
The image of C is C
C→C
A reflection in the line k is indicated in symbols as rk (k is written as a
subscript)
rk (A) = A’
rk (B) = B’
rk (C) = C
Line Reflections in Coordinate Geometry
1. Under a reflection in the y-axis:
P(x, y) → P’(-x, y)
Or ry-axis(x, y) = (-x, y)
2. Under a reflection in the x-axis:
P(x, y) → P’(x, -y)
Or rx-axis(x, y) = (x, -y)
3. Under a reflection in the line y = x:
P(x, y) → P’(y, x)
Or ry=x(x, y) = (y, x)
4. Under a reflection in the line y = -x:
P(x, y) → P’(-y, -x)
Or ry=-x(x, y) = (-y, -x)
Properties under a line Reflection:
1. Distance is preserved – each segment and its image are equal in
length
2. Angle measure is preserved – each angle and its image are equal in
measure
3. Parallelism is preserved – if two lines are parallel, then their images
will be parallel lines also
4. Collinearity is preserved – if 3 or more points lie on a straight line,
their images will also lie on a straight line
5. A midpoint is preserved – given 3 points such that one is the midpoint
of the other two, their images will be related in the same way
Ex: The vertices of ∆ABC are A(3, 0), B(3, 6), C(0, 6). What are the
coordinates of A’, B’ and C’ when ∆ABC is reflected in:
a) the x-axis
A’(3, 0)
B’(3, -6)
C’(0, -6)
b) the y-axis
A’(-3, 0)
B’(-3, 6)
C’(0, 6)
Are there any fixed points?
c) the line y = x
A’(0, 3)
B’(6, 3)
C’(6, 0)
5. Under a reflection in the line y = k:
ry=k(x, y) → (x, 2k - y)
(the x-value remains the same)
Ex: y = 2
If A(4, 1), what is the reflection under the line y = 2?
ry=k (x, y) → (x, 2k - y)
ry=2 (4, 1) → (4, [2(2) – 1]) → (4, 3)
A’(4, 3)
Under a reflection in the line x = h:
rx=h(x, y) → (2h - x, y)
(the y-value remains the same)
Ex: x = 4
If B(1, 4), what is the reflection under the line x = 4?
rx=h(x, y) → (2h - x, y)
rx=4 (1, 4) → ([2(4) – 1], 4) → (7, 4)
B’(7, 4)