[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:

• 1. Rotate 90° counter clockwise

  2. Reflect across x-axis

or

• 1. Rotate 90° counter clockwise

  2. 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

1. Reflect across x axis

2. Reflect across y axis

or

1. Reflect across y axis

2. Reflect across x axis

or

1. Rotate 180° clockwise

or

1. Rotate 180° counterclockwise

Example 2

What transformations can move ABCD onto "other"

There are 2 answers

1. Rotate 270° clockwise

or

1. Rotate 90° counterclockwise