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]. 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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.