Friday, December 22, 2006

Geometry - 7.1 + 7.2 guided notes

Geometry Unit 7 sec 7.1 and 7.2

Rigid Motion in a Plane/Reflections

Guided Notes:

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.

1. Butterfly

2. Leaf

3. Architecture

4. Automobiles

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?

(sometimes the lines are horizontal (MOM) and other times they are vertical (BIKED)