Point Reflection
Here, ∆ABC is reflected through point p and its image ∆A'B'C'
is formed.
Step 1) From each vertex of ABC, a segment is drawn
through p to its image such that the distance from
the vertex to point p is equal to the distance from
point p to its image.
Step 2) The images are connected to form ∆A'B'C' .
Which is the reflection of ∆ABC through point p.
A reflection in a point p – is a transformation of the plane such that:
1) The image of the fixed point P is P
2) For all other points, the image of K is K' where P is the
midpoint of KK ' .
In symbols:
Rp(A) = A' under the reflection in point P, the image of A is A'
Rp(B) = B' under the reflection in point P, the image of B is B'
Rp(C) = C' under the reflection in point P, the image of C is C'
The most common point refection in Coordinate Geometry is a
reflection in the origin (0,0)
Under a reflection in point 0, the origin,
P(x, y) → P'(-x, -y)
Or R0(x, y) = (-x, -y)
The properties preserved under a point reflection are the same as
those for line reflection (distance, angle measure, parallelism,
collinearity and midpoints)
To Find reflections of any point:
Use the midpoint formula to find the image
Ex 1) Given A(5, 5) and P(3, 2). When A is reflected in point p
find the coordinates of A'.
m(
x1  x2 y 2  y1
,
)
2
2
 5  x2 
3

 2 
 5  y2 
2

 2 
6 = 5 + x2 x2 = 1
4 = 5 + y2
y2 = -1
A' = (1, -1)
Ex 2) Given A(6, 3) and A'(2, -2) find the point of reflection
("midpoint").
Ex 3)
a) Using the rule (x, y) → (-x, -y) find the images of A(2, 1),
B(4, 5), C(-1, 3), namely A', B' and C'
b) On one set of axes, draw ∆ABC and ∆A'B'C'
c) Find the coordinates of M, the midpoint of AB .
d) Using AB and M and their images, show that a midpoint is
preserved under this transformation.
Ans: a) A'(-2, -1), B'(-4, -5), C'(1, -3)
c) (3, 3)
d) image of (3, 3) is (-3, -3)
midpoint of A' B' is (-3, -3)