Unit 4, Activity 1, Vocabulary Self-Awareness Word/Phrase + Definition/Rule Example rigid (rigid motion) non-rigid (non-rigid motion) orientation isometry reflection line of reflection translation rotation center of rotation Blackline Masters, Geometry Page 4-1
Unit 4, Activity 1, Vocabulary Self-Awareness degree of rotation clockwise counterclockwise dilation center of dilation scale factor similarity composite glide reflection Procedure: 1. Examine the list of words/phrases in the first column. 2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study. 3. Place a next to any words/phrases for which you can write either a definition or an example, but not both. 4. Put a next to words/phrases that are new to you. This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil. Blackline Masters, Geometry Page 4-2
Unit 4, Activity 1, Vocabulary Self-Awareness with Answers Word/Phrase + Definition/Rule Example A correspondence between two sets of points such that each point in the has a unique and that each point in the has exactly one ; a change in size, orientation, or position of a figure in space. The original object that is to be transformed. rigid (rigid motion) non-rigid (non-rigid motion) orientation isometry reflection line of reflection The copy of the object that has been transformed. A that preserves measurements of segments and angles; also called an isometry (see below). A that does not preserve measures of segments and angles; the shape of the may not be preserved either. The location (position and angle) of an object in space in relation to a set of reference axes. A that preserves measurements and more specifically distances between points; a that preserves distances is also bound to preserve angle measures; a congruence. A in which each point in the has an that is the same distance from the line of reflection (see below). For a point on the line of reflection, the is itself; aka flip. The perpendicular bisector of the segment joining each point () and its. Blackline Masters, Geometry Page 4-3
Unit 4, Activity 1, Vocabulary Self-Awareness with Answers translation rotation center of rotation angle (degree) of rotation clockwise counterclockwise dilation center of dilation scale factor A which moves an object a fixed distance in a fixed direction; a composite of two reflections over parallel lines; aka slide. A that turns a figure about a fixed point called the center of rotation; a composite of two reflections over intersecting lines; aka turn. A fixed point about which a figure is rotated; the point where two intersecting lines of rotation meet?? Rays drawn from the center of rotation to a point on the pre and its form the angle of rotation (measured in degrees). Rotation of an object to the right indicated by a negative angle of rotation. Rotation of an object to the left indicated by a positive angle of rotation. A that produces an that is the same shape as the but is a different size; a stretch or shrink of the. A fixed point in the plane about which all points are expanded (stretched) or contracted (shrunk). The ratio by which an object is enlarged or reduced; if greater than 1 the is an enlargement; if between 0 and 1 the dilation is a reduction; if the scale factor equals 1, the figures are congruent. Blackline Masters, Geometry Page 4-4
Unit 4, Activity 1, Vocabulary Self-Awareness with Answers similarity composite glide reflection A that is the composite of dilations and/or reflections; a non-rigid ; the shape of the is preserved but the size is changed. The result of two or more successive s. A type of composite where a figure is reflected then translated. Procedure: 1. Examine the list of words/phrases in the first column. 2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study. 3. Place a next to any words/phrases for which you can write either a definition or an example, but not both. 4. Put a next to words/phrases that are new to you. This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil. Blackline Masters, Geometry Page 4-5
Unit 4, Activity 3, A Basic Look at Transformations Trace the following polygons on the patty paper or tracing paper given to you. Blackline Masters, Geometry Page 4-6
Unit 4, Activity 4, What Are Transformations? What Are Transformations? When learning about s, one might first look at the parts of the word. Transformation can be separated into the prefix trans- and the word formation. The prefix transmeans changing thoroughly and formation means the act of giving or taking form, shape, or existence. Taken together, a can be described as the act of changing a form or shape. Specifically, in geometry, a may change the position, orientation, or size of a figure in the plane. Look at the s below. The figure drawn with the solid lines is called the, or the original object that is being transformed. The figure drawn with dashed lines is called the, or the copy of the object that has been transformed. Figure 1 Figure 2 The most basic is a reflection. A reflection can be easily described as a flip, however, that is not the most accurate definition. A reflection is a in which each point in the has an that is the same distance from the line of reflection. The line of reflection is the perpendicular bisector of the segment joining each point on the pre with its corresponding point on the. Look at the example below. The is labeled A B C D (read A prime, B prime, C prime, D prime the use of the apostrophe on the letter is universally accepted to show that a figure is the of a ). D A P A D C B m B C Figure 3 Line m is the line of reflection in the figure and P is the point where line m intersects the segment joining A and A. Using a ruler, measure the distance from A to A. Now, measure the distance from A to P and the distance from P to A. You should notice that AP and PA are equal. Use a protractor to measure the angles formed at P. You should see that the angles all measure 90 Blackline Masters, Geometry Page 4-7
Unit 4, Activity 4, What Are Transformations? degrees. How do those measurements relate to the definition of the line of reflection given earlier? Another basic is a translation, or slide. When a translation is performed, the is moved a fixed distance in a fixed direction. The directions for performing a translation could state to move the 5 inches to the right in which each point on the is moved 5 inches to the right of the location to form the. Translations can also be thought of as the composition of two reflections over parallel lines. Look at the example below. m Figure 4 n Notice there are two reflections. Lines m and n are parallel. The resulting has the same orientation as the, but has been moved to the right by 10 cm. A question to think about: does the distance of the translation have any relationship to the distance between the parallel lines? A third is called a rotation. This may also be referred to as a turn. A rotation is a that turns a figure about a fixed point, called the center of rotation, through a fixed angle of rotation (measured in degrees). The path the figure follows during the rotation would form a circle around the center of rotation if the figure were rotated 360 degrees. A rotation can be performed with any degree measure and can be considered a clockwise rotation or a counterclockwise rotation. A clockwise rotation will turn a figure to the right around the center of rotation while a counterclockwise rotation will turn a figure to the left around the center of rotation. All positive degree measures are assumed to indicate a counterclockwise rotation, while all negative degree measures are assumed to indicate a clockwise rotation. X Figure 5 Blackline Masters, Geometry Page 4-8
Unit 4, Activity 4, What Are Transformations? In Figure 5 above, X is the center of rotation. The angle of rotation is formed by drawing a segment from one point on the to the center of rotation then drawing the required angle using the center of rotation (X) as the vertex. The angle used in this figure is 90 clockwise, or -90. Question to think about: What would happen to the if the center of rotation was moved but the angle of rotation remained the same? A rotation can also be defined as a composite of two or more reflections over intersecting lines. Consider the example below. m n Figure 6 The intersection of lines m and n becomes the center of rotation. These lines happen to be perpendicular. Notice how the has been rotated in a counterclockwise direction around the center (point of intersection). How could you determine what the angle of rotation is for this diagram? Go back to the definition of rotation discussed earlier for some ideas. The three s discussed so far have one thing in common. If you look at all of the s and compare them to their corresponding s, you will notice that the measures of the segments and angles have not changed (go ahead measure them if you wish!). Since the s have the same shape and are the same size as the s, they are congruent. Each of these s is called an isometry. An isometry is a that preserves measurements of segments and angles and therefore produces an congruent to its. A that is an isometry is also sometimes referred to as a rigid motion or rigid. In geometry, there is one more important. A dilation is a that produces an that is the same shape as the but is a different size. Sometimes they are referred to as a stretch or shrink (also called an enlargement or reduction). Each dilation is focused at the center of dilation, or a fixed point about which all points are enlarged or reduced. How much the figure is enlarged or reduced depends upon the scale factor, the ratio by which an object is enlarged or reduced. If the scale factor is greater than 1, the is an enlargement of the. If the scale factor is between 0 and 1, the is a reduction of the. B B Figure 7 A E D C E D Blackline Masters, Geometry Page 4-9 C
Unit 4, Activity 4, What Are Transformations? In Figure 7 above, the center of dilation is A. The measure of segment AB is 2 times the measure of segment AB. Therefore, the A B C D E is an enlargement of the ABCDE, and the scale factor is 2. Notice, the measures of the corresponding segments are not equal, however the measures of the corresponding angles are (you can verify this by using your protractor). Therefore, dilation is not an isometry. Dilation is a non-rigid, or a non-rigid motion. Because the corresponding angles have the same measure and the corresponding sides are proportional, these figures are similar which means dilation is a similarity. Blackline Masters, Geometry Page 4-10