Compositions of Isometries
Compositions of Isometries
Standard
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.
"Compositions" are like a collection of things
"Compositions" are like a collection of things
"Isometries" are translations, reflections, and rotations
"Isometries" are translations, reflections, and rotations
So compositions of isometries are a collection of translations, reflections, and/or rotations.
So compositions of isometries are a collection of translations, reflections, and/or rotations.
- This means a question might ask you to to translate, then rotate, then reflect, or any combination of isometries.
Composition Notation
Composition Notation
The circle (०) means "composed". So the following says, "Rm composed of Rl of ABC is A double prime..." etc.
The circle (०) means "composed". So the following says, "Rm composed of Rl of ABC is A double prime..." etc.
It might be easier to re-write compositions like this:
It might be easier to re-write compositions like this:
Step 1: PEMDAS. Do what's in the parentheses first! So do the Reflection across line l first.
Step 1: PEMDAS. Do what's in the parentheses first! So do the Reflection across line l first.
Step 2: Now do the Reflection across line m.
Step 2: Now do the Reflection across line m.
Example
Example
What does this mean? (Ry=-2 ० T<1,0>)(ABC)
- This can be re-written as Ry=-2(T<1,0>(ABC)).
- So we're applying the Translation of ABC first. ABC is moved 1 unit to the right and 0 units up/down.
- Then we do the Reflection across the line y = -2.