ACTIVITY 1 Exercise Match the figures (a,b and c) with the corresponding type of isometry

Transcription

1 ACTIVITY 1 Exercise Match the figures (a,b and c) with the corresponding type of isometry a) b) c) 1. Rotation 2. Translation 3. Reflection (mirror isometry) Figure Isometry Exercise Determine the number of lines of symmetry N N N

2 ACTIVITY 2 Task: Identify the type of isometry and describe it completely Hints: Connect the corresponding points (A-A, B-B.). What do you see? Referring to the figure above, mark the correct answer (s) for each column Isometry Described By Type Fixed Point Centre Glide Angle ( ) Zero Reflection Direct Line One Rotation Opposite Vector Infinite Translation (Moduls Cm)

3 ACTIVITY 3 Task: Identify the type of isometry and describe it completely Hints: Draw the segments AA, BB.and draw their axis. What do you see? Referring to the figure above, mark the correct answer (s) for each column Isometry Described By Type Fixed Point Centre Glide Angle ( ) Zero Reflection Direct Line One Rotation Opposite Vector Infinite Translation (Moduls Cm)

4 ACTIVITY 4 Task: Identify the type of isometry and describe it completely Hints: Draw the segments AA, BB.and draw their axis. What do you see? Referring to the figure above, mark the correct answer (s) for each column Isometry Described By Type Fixed Point Centre Glide Angle ( ) Zero Reflection Direct Line One Rotation Opposite Vector Infinite Translation (Moduls Cm)

5 LABORATORY- ACTIVITY 5 Owen sayd that since a reflection in the x-axis followed by a reflection in y-axis has the same result as a reflection in the y-axis followed by a reflection in x-axis, the composition of line reflections is a commutative operation. Do you agree with Owen? Justify your answer using Geogebra. LABORATORY- ACTIVITY 6 A reflection in the line y=x followed by a rotation of about the origin in equivalent to what single transformation? A rotation of about the origin followed by another rotation of about the origin is equivalent to what single transformation? Draw a triangle and justify your answer using Geogebra. LABORATORY- ACTIVITY 7 Work with a partner a) Take a piece of paper and fold it in half Draw a shape on the folden paper, starting and finishing at the fold line. Cut out your shape and open it out Your shape has one line of symmetry b) Make other shapes with one line of symmetry c) By folding your paper twice, make shapes with two lines of symmetry d) How can you make shapes with four lines of symmetry? ACTIVITY 8 Goals:Transformations and fixed points Answer the following questions a) For what transformation or transformations are there no fixed points? b) For what transformation or transformations is there exactly one fixed point? c) For what transformation or transformations are there infinitely many fixed points?

6 LABORATORY- ACTIVITY 9 Complete the theorem Theorem 1 : the product of any two reflections in the plane is a translation or rotation. In particular,if the mirror lines Then their products is a translation by vector v with direction and modulus. If the mirror lines their product is a rotation through angle with center Hint:Begin with something like the following:

7 FINAL TEST 1. Which of the following is not an isometry a) A rotation b) A reflection c) A translation d) A dilation 2. Which of the following statements is not true? a) A reflection in a line is congruent to the original figure b) Corresponding sides of a figure and its reflection in a line parallel c) Corresponding sides of a figure and its reflection in a line are congruent d) The line of symmetry bisects a segment connecting corresponding points of a figure and its reflection e) The line of symmetry is perpendicular to a segment connecting corresponding points of a figure and its reflection. 3. A quadrilateral that is central symmetric with respect to a point is always: a) A rectangle b) A rhombus c) A parallelogram d) A square e) A trapezoid 4. How many different isometries transform a regular pentagon in itself? 5. The direct isometries are: a) Translation and symmetries b) Rotations and symmetries c) Rotations and translations d) Rotations e) Other 6.The opposite isometries are: a) Translation and rotations b) Symmetries and translations c) Glides and symmetries d) Glides e) Other

8 7. Complete this tableby listing the letters of the alphabet that match the description of symmetry a) One line of symmetry A,T,.. b) Two lines of symmetry H,.. c) Rotational symmetry of order 2 N,. d) Rotational symmetry of order 4 H,. e) More than 4 lines of symmetry. f) No lines of symmetry,no rotational symmetry F,G,.. g) Two lines of symmetry,rotational symmetry of order The diagram consist of nine congruent rectangles.under a translation,the image of A is G. Find the image of each of the given points under the same translation. A B C D E F G H I J K L M N O P a) J b) B c) I d) F e) E 9. Let Q be the image of P under a clockwise rotation of 90 about the origin and R b the image of Q under a clockwise rotation of 90 about the origin. For what two different transformations is R the image of P? 10. On a graph paper, locate the points A( 3;3), B ( 3;7) and C ( -2;7). Draw ΔABC. Draw ΔA B C, the image of ΔABC under a reflection in the origin and write the coordinates of its vertices. Draw ΔA B C the image of ΔA B C under a reflection in the y-axis and write the coordinates of its vertices. Under what single transformation is ΔA B C the image of ΔABC?

9 Summary A transformation is a one- to -one correspondence between two sets of points, S and S, when every point in S corresponds to one and only one point in S called image, and every point in S is the image of one and only one point in S called preimage. An isometry is a transformation that preserves the distance, angles,orthogonality and parallelism. The isometries of the plane are:translations, Rotations,Reflections and Glide Reflections. A translation is a transformation in a plane that moves every point in the plane the same distance in the same direction.(no fixed point-identity if the vector is null) A rotation is a transformation of a plane about a fixed point P through an angle of d degrees such that: 1) for A, a point that is not the fixed point, if the image of A is A,then PA=PA and m<apa =d A reflection in line k is a transformation in a plane such that:1) if point A is not on k, then the image of A is A where k is perpendicular bisector of AA 2) if A is on k,the image of A is A. A figure has line symmetry when the figure is its own image under a line reflection. A point reflection in P is a transformation plane such that: 1)If point A is not point P, then the image of A is A where P the midpoint of AA 2) The point P is its own image. A glide reflection is a composition of transformations of the plane that consists of a line reflection and a translation in the direction of the line of Reflection in either order An isometry of the plane is the composition of at most three reflections; thus the translations (compositions of two reflections with parallel axis) and the rotations (compositions of two reflections with non parallel axis) preserve the orientation,and the reflections and glide reflections change it (a glide reflections is the composition of three reflections with at least two not parallel axis).