Rigid Motion in a Plane/Reflections
I) Rigid Motion in a Plane sec 7.1
A) Transformation: Movement of an original figure (preimage) onto a new figure ( image).
In a figure of two triangles, ABC and A'B'C', the preimage is triangle ABC and the image is triangle A'B'C' (stated triangle A prime, B prime, C prime)
Triangle ABC will fit exactly on top of the other by doing a transformation of the figure.
Vertex A maps to Vertex A', vertex B maps to Vertex B' and vertex C maps to vertex C'
B) Isometry: A transformation that preserves lengths, angle measures, parallel lines, and distance between points. They are called rigid transformations.
II) Reflections sec 7.2
C) Line of Reflection: the line that a figure is reflected over - if we have a triangle reflect over a line, the image can be fitted exactly over the preimage by folding the paper on the line of reflection
Now let's look at specific line reflections
1. Reflection in x-axis: (x,y) maps to (x,-y)
Example: (4 , 3) becomes (4 , -3)
2. Reflection in y-axis: (x,y) maps to (-x,y)
Example: (4 , 3) becomes (-4 , 3)
3. Reflection in y = x: (x,y) maps to (y,x )
Example: (4 , 3) becomes (3 , 4)4. Reflection in y = -x: (x, y) maps to (-y, -x)
D) Line of Symmetry: A figure allows a copy of a figure to be mapped onto itself.
In nature, art, and in industry, we find many forms that contain a line of reflection.
E. Axis of Symmetry - when the figure is its own image under a reflection in a line.
Example: An Isosceles triangle - draw a line from the vertex of the angle that is not a base angle perpendicular to the base side (non-congruent side). This line is the axis of symmetry because it divides the triangle into 2 congruent triangles.
Triangle ABC is an isosceles triangle with sides AB = BC and angle A = angle C. Draw a perpendicular line from vertex A through AC and where they intersect label this point D. Line segment BD is the axis of symmetry and the reflection line so triangle ABD = triangle CBD.
Example 2: Can you think of any letters of the alphabet that have line symmetry?
A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, Y
How about any words?
MOM, BIKE, HIKED, CHECK, BOB, DEED, RADAR,
(sometimes the lines are horizontal (MOM) and other times they are vertical (BIKED)