[G.CO.5] Congruence #5
Objective
Common Core Text:
[G.CO.5] Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Said Differently:
Rotate, reflect, and translate using
graph paper
tracing paper
geometry software (GeoGebra)
Given two figures, determine which transformations can move one figure onto the other
Example
How can you transform rectangle ABCD onto "Other"?
Answer:
Rotate 90° counter clockwise
Reflect across x-axis
or
Rotate 90° counter clockwise
Translate down 5 spaces
(or many other possible answers)
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Explanation
Now that you've studied transformations, it's time to actually perform them.
Translations
Translate the shape by the vector <1,-3>
The vector <1,-3> is a lot like an ordered pair. It means that every point must move +1 in the x direction and -3 in the y direction.
Using Graph Paper
Translations with graph paper are pretty simple. Just remember that each point must move according to the vector. Moving each point right 1 and down 3, we get:
Figure A'B'C'D' is the translated shape.
Using Tracing Paper
The following tutorial demonstrates how to use tracing paper to translate a shape
Using Geometry Software (GeoGebra)
The following tutorial demonstrates how to translate a shape using GeoGebra
Reflections
Reflect the shape across the line y = -1
Using Graph Paper
First draw our line of reflection, given as y = -1
Now begin drawing your new points.
A is 2 spaces above the line of reflection, so A' must be 2 spaces below
B is 3 spaces above the line of reflection, so B' must be 3 spaces below
Continue drawing points and create your image
A'B'C'D' is our reflected shape
Using Tracing Paper
The following tutorial demonstrates how to use tracing paper to reflect a shape
Using Geometry Software (GeoGebra)
The following tutorial demonstrates how to reflect a shape using GeoGebra
Rotations
Rotate shape ABCD 90° clockwise about the origin
Using Graph Paper
Rotations are the hardest transformation to perform with just a piece of graph paper. You've really gotta use your imagination to see that shape spinning around in a circle. It takes a lot of practice.
For this example, we need to rotate 90°. Since every quadrant in a coordinate system is 90°, that means I need to move the shape over one quadrant. Since it's clockwise about the origin, I know its going to the right.
When rotating 90°, I like to think like this: the distances from the x-axis and y-axis switch. What do I mean by that? Lets take point D for example. Now, it's 1 space above the x-axis. So when I rotate it, it should be 1 space to the right of the y-axis. It's 5 spaces to the left of the y-axis, so when I rotate it will be 5 spaces above the x-axis.
If we make a table, it looks like this. Notice how the original point and its image switch x and y coordinates (but whether it will be positive or negative you have to imagine for yourself)
Rotating the rest of the points, we get our answer
Using Tracing Paper
The following tutorial demonstrates how to use tracing paper to rotate a shape. Using tracing paper is a great way understand how rotations work.
Using Geometry Software (GeoGebra)
The following tutorial demonstrates how to rotate a shape using GeoGebra
Identifying transformations
This just takes imagination and practice.
Example 1
What transformations can move ABCD onto "other"
There are 4 answers
Reflect across x axis
Reflect across y axis
or
Reflect across y axis
Reflect across x axis
or
Rotate 180° clockwise
or
Rotate 180° counterclockwise
Example 2
What transformations can move ABCD onto "other"
There are 2 answers
Rotate 270° clockwise
or
Rotate 90° counterclockwise