Geometry as Transforming Shapes, Exams of Geometry

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CHAPTER 9

Geometry as

Transforming

Shapes

SECTION 9.

Congruence Transformations

Translations

Translations In geometry, a translation involves moving (sliding) a shape, without rotating or flipping it. The shape still looks exactly the same, just in a different place. A translation is a special kind of congruence transformation These design show translations.

Understanding Translations One common response is to say that we slide the triangle from one location to the other. The mathematical word for slide is translation , and we talk about a figure and its image. Various Ways to Describe Translations: 1.Taxicab 2.New Notation 3.Specify the Distance and the angle 4.Vectors

CAT to C’A’T’ Some of the perspectives are easier to describe if we draw the two triangles on graph paper. cont’d

Translations: New Notation Invent New Notation We can use notation to express this same idea more succinctly: ( x , y ) → ( x + 3, y + 4) This notation gives the directions “go 3 units to the right and 4 units toward the top of the page” very succinctly.

Translations: Distance and Angle Distance and Angle The figure has been moved a distance of 5 units at a 54-degree angle from the x -axis. That is, C and C  are 5 units apart; the distance between any point on triangle CAT and the corresponding point on triangle C  A  T  is 5 units. Similarly, the angle formed by the rays CC  and CT is equal to approximately 54 degrees.

Properties of Translations

Properties of Translations Now let us determine some of the properties of translations. Earlier we know that two points determine a line. Therefore, if we connect each vertex in the triangle TAR to its image, T  A  R , we have three line segments: TT , AA , RR  The three line segments are all congruent (same length) and all parallel (same direction).

Translating using a Vector We can draw four lines, each going through a vertex, that are the same length and parallel to the translation vector. If we copy the length of the vector using a compass, we can quickly mark off the same lengths on the lines to determine the vertices of M  A  T  H .

Reflections

Understanding Reflections Below is a trapezoid that has been reflected (flipped) in three different ways. In each case, bold lines denote the original figure and dotted lines denote the flipped image.

Properties of Reflections True for all reflections: if we connect any point on the original figure with the corresponding point on the reflected figure, the line of reflection is the perpendicular bisector of that line segment. That is, A  X = XA , and and line l are perpendicular.