Isometries and Congruence

Transcription

1 Honors Geometr Section.1 Name: Date: Period: Isometries and Congruence transformation of a geometric figure is a change in its position, shape, or size.. The original figure is called the preimage. The transformed figure is the image. n isometr is a transformation in which the image and preimage are congruent. ll of the angles are congruent ll of the sides are congruent Translations: Figure JKL is translated units down to form image J K L. a) Based on the diagram above, what do ou think it means to translate a figure? b) What is the difference between a preimage and an image? c) Using our protractor, measure the following angles. d) What patterns do ou notice in the table on the right? ngle J K L J K L Measurement e) Find the length of each line segment in the table below. f) How does the length of compare to? Line Segment Measurement

2 g) Fill in the following table: Point J J K K L L Coordinates h) What meaningful observations can ou make about the corresponding coordinates J and J? K and K? L and L? i) Can ou come up with a formula in terms of (x, )? The formula is sometimes referred to as the translation rule. j) Based on our two tables above, what is the relationship between a preimage and its image under a translation? What properties must sta the same under a translation? Problem 1: Figure BCD has the vertices (-5, 3), B(-2, 4), C(-1, 1) and D(-3, 1). Sketch the preimage BCD and its image B C D after the translation rule: ( x, ) ( x 5, 3) x

3 VECTOR NOTTION: nother wa to describe a translation is b using a vector. vector is a quantit that has both direction and magnitude, or size. vector is represented in the coordinate plane b an arrow drawn from one point to another. vector is a directed line segment. x The initial point, or starting point, of the vector is. The terminal point, or ending point, of the vector is B. The component form of a vector combines the horizontal and vertical components. The component form of B is, B 5 units up vertical component units right horizontal component Example: The vertices of BC are (-1, 3), B(3, 5) and C(2, 0). Translate BC using the vector, 3. Label the image ' B' C' C B x figure is said to have clockwise orientation if its vertices when read, B, C are pictured in clockwise orientation and a figure is said to have counterclockwise orientation if its vertices are pictured in counterclockwise orientation.

4 reflection is a transformation that uses a line like a mirror to reflect an image. The mirror line is called the line of reflection, m. m B C C B The reflection of a triangle across the line x = 4 is shown below. The line x = 4 is the line of reflection. a. Compare the distance from vertex B to the line of reflection with the distance from vertex B to the same line. b. Compare the distances of C and C to the line of reflection. c. Do the same for D and D. d. What do ou notice about the distances of each vertex to the line of reflection? e. Line segment corresponds with line segment. What do ou notice about the lengths of the corresponding sides of the triangles? f. What properties remain the same under a reflection? What properties change? g. Problem 2: Figure BCD has the vertices (-5, 3), B(-2, 4), C(-1, 1) and D(-3, 1). Draw the reflection of figure BCD across the -axis. Label this image B C D. Then, draw the reflection of figure B C D across the x-axis. Label this new figure B C D

5 Coordinate Rules for Reflections If (x, ) is reflected in the x-axis, its image is the point (x, -). If (x, ) is reflected in the -axis, its image is the point (-x, ). If (x, ) is reflected in the line = x, its image is the point (, x). If (x, ) is reflected in the line = -x, its image is the point (-, -x). Problem3: Figure BCD has the vertices (-5, 3), B(-2, 4), C(-1, 1) and D(-3, 1). Reflect the figure over the line = x. Label this figure B C D. Then, reflect the figure over the line = -x. Label this figure B C D B D C x

6 Rotations: rotation is a transformation in which a figure is turned around a fixed point called the center of rotation. Ras drawn from the center of rotation to a point and its image form the angle of rotation. The angle of rotation is a directed angle where one side is the initial side and a second side the terminal side. Investigation: Flag 1 1. Rotate Flag 1 0 degrees clockwise around the origin. Label the rotated image as Flag Rotate Flag 1 10 degrees clockwise around the origin. Label the rotated image as Flag Rotate Flag 1 20 degrees clockwise around the origin. Label the rotated image as Flag Rotate Flag 1 30 degrees clockwise around the origin. Write down an observations about what occurs. Example 1: Figure BCD undergoes a rotation of 0 degrees clockwise.

7 Which quadrant will the arrow end up in? i. Quadrant I ii. Quadrant II iii. Quadrant III iv. Quadrant IV Which direction will the arrow be pointing? i. iii. ii. iv. Example 2: Draw the rotation of triangle TUV 0 degrees clockwise.

8 Now You Tr: Draw the rotation of triangle GHI 10 degrees counterclockwise. What are the coordinates of G? rotation about a point P through x degrees is a transformation that maps ever point Q in the plane to a point Q, so that the following properties are true: B If Q is not point P, then QP = Q P and m QPQ ' x. If Q is point P, then Q = Q. rotation can be clockwise or counterclockwise depending upon the degree of rotation. B C P C

9 Coordinate Rules for Rotations about the Origin. When a point (a, b) is rotated counterclockwise about the origin, the following are true: For a rotation of 0 (-20 o ) ( a, b) ( b, a) For a rotation of 10 (-10 o ) ( a, b) ( a, b) For a rotation of 20 (-0 o ) ( a, b) ( b, a). Example: Figure BCD has the vertices (-5, 3), B(-2, 4), C(-1, 1) and D(-3, 1). Rotate the figure -0 about the origin. B D C x Glide Reflections: transformation of a glide and a reflection can be performed one after the other to produce a transformation know as a glide reflection. glide reflection is a transformation in which ever point P is mapped onto a point P b the following steps: translation maps P onto P. reflection in a line k parallel to the direction of the translation maps P onto P. s long as the line of reflection is parallel to the direction of the translation, it does not matter whether ou glide first and the reflect, or reflect first and then glide.

10 Example: Figure BCD has the vertices (-5, 3), B(-2, 4), C(-1, 1) and D(-3, 1). Sketch the image of figure BCD after the glide reflection using translation ( x, ) ( x, ) and reflection in the x-axis.