IM Geo.6.1 Lesson: Rigid Transformations in the Plane

How far is point A from point B?[br]
What transformations will take point A to point B?
Rotate Figure H clockwise using center (2,0) by 90 degrees. Translate the image by the directed line segment from (2,0) to (3,-4). Label the result R.
Rotate Figure H across the y-axis. Rotate the image counterclockwise using center (0,0) by 90 degrees. Label the result L.
Calculate the length of each side in triangles [math]ABC[/math] and [math]DEF[/math].[br]
Calculate the measure of each angle in triangles  [math]ABC[/math] and [math]DEF[/math].
Because the triangles are congruent, there must be a sequence of rigid motions that takes one to the other. Find a sequence of rigid motions that takes triangle  [math]ABC[/math] to triangle [math]DEF[/math].
The triangles are congruent. How do you know this is true?
What single transformation would take triangle [math]ABC[/math] to triangle [math]DEF[/math]?

IM Geo.6.1 Practice: Rigid Transformations in the Plane

Reflect triangle ABC over the line x=-3. Translate the image by the directed line segment from (0,0) to (4,1).
What are the coordinates of the vertices in the final image?
Three line segments form the letter N. Rotate the letter N counterclockwise around the midpoint of segment BC by 180 degrees.
Describe the result.
[size=150]Triangle [math]ABC[/math] has coordinates [math]A=(1,3),B=(2,0),[/math] and [math]C=(4,1)[/math]. The image of this triangle after a sequence of transformations is triangle [math]A'B'C'[/math] where [math]A'=(\text{-}5,\text{-}3),B'=(\text{-}4,0)[/math] and C'=[math](\text{-}2,\text{-}1)[/math]. [br][img]data:image/png;base64,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[/img][br][/size]Write a sequence of transformations that takes triangle [math]ABC[/math] to triangle [math]A'B'C'[/math].
[size=150]Prove triangle [math]ABC[/math] below is congruent to triangle [math]DEF[/math].[/size]
[size=150]The density of water is 1 gram per cm³. An object floats in water if its density is less than water’s density, and it sinks if its density is greater than water’s. Will a 1.17 gram diamond in the shape of a pyramid whose base has area 2 cm[math]^2[/math] and whose height is 0.5 centimeters sink or float? Explain your reasoning. [/size]
An oblique cylinder with a base of radius 2 units is shown.
[size=150]The top of the cylinder can be obtained by translating the base by the directed line segment [math]AB[/math] which has length 16 units. The segment [math]AB[/math] forms a [math]30^{\circ}[/math] angle with the plane of the base. [/size][br][br]What is the volume of the cylinder?
This design began from the construction of an equilateral triangle.
Record at least 3 rigid transformations (rotation, reflection, translation) you see in the design.

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