Defining congruence through rigid transformations

The Defining congruence through rigid transformations exercise appears under the High school geometry Math Mission. This exercise uses a manipulative to explore geometric transformations and better understand how they are connected with the congruence relation.

Types of Problems

There are two types of problems in this exercise:

1. Perform transformations to move the polygon: This problem provides a plane with two polygons drawn on it. The student is asked to determine a sequence of transformations that will map one polygon onto the other, if possible, and determine if the polygons are congruent.

   

   Perform transformations to move the polygon

2. Perform the transformations on the polygon: This problem provides a polygon and a set of transformations. The student is asked to follow the sequence of transformations and determine if the shapes at the beginning and end are congruent.

   

   Perform the transformations on the polygon

Strategies

Knowledge of isometries and familiarity with the Cartesian coordinate plane are encouraged to ensure success on this exercise.

1. On the Perform transformations to move polygon problem, one way to effective find a correct sequence of transformations is to translate a point in one polygon to a corresponding point on the other one. Once this is completed, reflecting and rotating as needed may be clearer.
2. The rotations appear to always be multiples of ninety degrees.
3. On the Perform the transformations on the polygon problem, the transformations of rotation, reflection and translation are all isometries. Therefore using these preserves congruency.

Real-life Applications

1. Recognizing when shapes are congruent, and conditions that ensure congruency, is an important skill in drafting and architecture.
2. Architects use lots of geometry when building bridges, roofs on houses, and other structures.
3. The ancient Egyptians from over 4000 years ago were very good at shapes and geometry. Every time the Nile burst its banks and flooded the planes, they had to use geometry to measure their gardens and fields all over again.