Chapter 9 Congruence, Symmetry, and Similarity Section 9.1 Transformations and Congruence

Congruence The concept of congruence we mentioned before with angles. The intuitive concept of congruent is if one shape can be “picked up” and “placed down” on another shape so that each shape exactly matches the other. This means that the two figures are exactly the same shape and size. The tricky issue is what is meant by “picked up” and “placed down”. The procedures that are used to “pick up” and “put down” shapes are called rigid transformations or isometries. The rigid transformations that can be done to a shape can be broken down into a combination of one of three specific types: a. Translation b. Reflection c. Rotation The resulting shape (really the new position of the object) is called the image under the transformation. Translations Translations “slide” the points of the object along a path or paths so that each point of the object is “slid” the exact same distance.

Here are some examples of translations: We say the green triangle is congruent to the black triangle and the orange pentagon is congruent to the black pentagon. Translations can also be conceptualized on a geoboard as illustrated below. The blue isosceles triangle is congruent to the black isosceles triangle. On the geoboard it has been translated 5 units to the right and 4 units down. The blue isosceles triangle is the image of the black isosceles triangle under the transformation.

Reflections A reflection uses a line like a “mirror” to reproduce a shape exactly so that the corresponding points of the shape are the exact same distance from the line but on opposite side of the line. We call this a reflection across the line. A B The reflection of the pentagon across C D The reflection of the triangle across On the geoboard the blue trapezoid is the reflection of the black trapezoid across the brown line. The two trapezoids are congruent.

Rotations A rotation “spins” or turns the shape. The is one point that does not change for a rotation and we call that the point which you rotate about. A The green triangle is a 90  clockwise rotation of the black triangle about the point A. The purple parallelogram is a 180  clockwise rotation about the point B. B C On this geoboard the green triangle is a 90  counterclockwise rotation of the black triangle about the point C. Sometimes we call C the center point of the rotation.