Key visualizations: Geometry

Geometry provides numerous opportunities for students to make and critique arguments, using precise language. Choosing appropriate tools is a key part of the exploration process that leads to viable arguments.

Connections to the TEKS []
This animation addresses the following Geometry TEKS:

(1) Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;
(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

(3) Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and rotation) and non-rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity). The student is expected to:
(B) determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane;
(C) identify the sequence of transformations that will carry a given pre-image onto an image on and off the coordinate plane;

(7) Similarity, proof, and trigonometry. The student uses the process skills in applying similarity to solve problems. The student is expected to:
(A) apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles;


About this animation []

In this animation, students use transformations to show two triangles are similar. Students can use Patty Paper or MIRAs to work in small groups to find a sequence of transformations that will map ∆ABD onto ∆DEF. The key is for students to recognize the need for a reflection and dilation. Once students have found a sequence of transformations, they should develop a justification to share with other small groups. As student groups share their justifications, they should be encouraged to use precise language about the transformations and similarity. Groups should provide critical feedback to other groups about their reasoning. As students share their reasoning, the point that the existence of such a sequence proves the triangles are similar should emerge. An extension question for students to consider following this experience is:

If you could map a shape onto another shape with a sequence of transformations that does not include a dilation, are the two shapes still similar?

The intent of this question is to consider congruence as a special case of similarity, namely where the scale factor is 1.



Transforming…
The reflection is necessary to change the orientation of △ABC to match that of △DEF. But, you need another transformation as well. Clear the image and try again.
Transforming…
A rotation is not needed in this case to change the orientation of △ABC. Clear the image and try again.
Transforming…
You are correct that a dilation is needed, but start by getting the orientation of △ABC the same as △DEF. Clear the image and try again.
Transforming…
Very good! You can start by reflecting △ABC to match the orientation of △DEF. Rays connecting corresponding points of the triangles intersect at the center of a dilation.
Select the transformations necessary to transform △ABC onto △DEF. Once you have made your selections, press Transform.
△ABC is similar to △DEF because △DEF can be mapped onto △ABC by a dilation and a reflection.