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Grade 9 MathUnit 8 – Similarity & Transformations
8.7 – Identifying Types of Symmetry on the Cartesian Plane
Investigation: Do the investigation below in groups of 3.
Plot these points on the coordinate grid below:
1. Join the points to form ΔABC.
2. Each of you chooses one of these transformations:
A translation 2 units right and 2 units down.
A rotation of 180 about vertex C. ̊ A reflection in a line through AB.
3. Draw the image for the transformation you chose on the grid above.
4. Record the coordinates of each vertex on the image.
Answer: Translation A ̍(3,1) B ̍(5,-1) C ̍(7,3)Rotation A ̍(9,7) B ̍(7,9) C ̍(5,5)Reflection A ̍(1,3) B ̍(3,1) C ̍(-1, -1)
5. Record any symmetry in the triangle and its image.
6. Compare with your group members the types of symmetry you each found. Did any show both rotational symmetry and line symmetry?Answer: Translation was also a reflection across the line through (3,1) & (5,2) Rotation is also a reflection across the line through (4,6) & (6,4)
Examine this grid.
What types of symmetry are shown?
Answer: Rectangle A has been rotated 180 about E(-1,2) to produce rectangle B. ̊Rectangle A has been reflected in the vertical line that passes thru x = -1.
Conclusion:When a shape and its transformation image are drawn, the resulting diagram may show:
No symmetry Line symmetry (reflection) Rotational symmetry Both line symmetry and rotational symmetry
Ex. 1 For each pair of rectangle ABCD and EFGH, determine whether they are related by symmetry, and if so, what kind.
a.
Answer: They are related by rotational symmetry of order 2, angle of rotation of 180 about (0,3). There is no line symmetry. ̊
b.
Answer: Related by line symmetry across the x-axis. Related by rotational symmetry of order 2, angle of rotation of 180 , about (-2.5, 0). ̊
c
Answer: Related by rotational symmetry of order 4, angle of rotation of 90 clockwise, about (-5,4). ̊
No line symmetry.
Ex. 2 Draw the image of rectangle ABCD after each transformation. Write the coordinates of each vertex of its image. Identify and describe the type of symmetry that results.
a. A rotation of 180 about the origin. ̊
Rotational symmetry of 180 about (0,0) ̊ No line symmetry
b. A reflection in the x-axis.
across the x-axis (or y = 0)
of 180 about (1,0) ̊
c. A translation 4 units right and 1 unit down. Rotational symmetry of 180 about (3,0) ̊ No line symmetry
Note, that when writing translations, there is a shorthand form. For example, when writing the translation 4 units right and 1 unit down, we can write this as R4, D1.
Ex. 3 Write the shorthand form for the translation 7 units left and 2 units up.
Answer: L7, U2
Ex. 4 Draw the image of the pentagon PQRST after each translation below. Label the vertices of the pentagon’s image and list their coordinates. If each diagram has symmetry, describe it.
a. A translation L2.
b. A translation L2, D3.
Assignment Do #1 – 15 (omit #14) p. 373Do Unit Review #1 – 19 p. 377
The unit test will be on _____________________________!
Grade 9 MathUnit 8 – Similarity & TransformationsStudent Copy
8.7 – Identifying Types of Symmetry on the Cartesian Plane
Investigation: Do the investigation below in groups of 3.
Plot these points on the coordinate grid below:
A (1,3) B (3,1) C (5,5)
1. Join the points to form ΔABC.
2. Each of you chooses one of these transformations:
A translation 2 units right and 2 units down.
A rotation of 180 about vertex C. ̊ A reflection in a line through AB.
3. Draw the image for the transformation you chose on the grid above.
4. Record the coordinates of each vertex on the image.
5. Record any symmetry in the triangle and its image.
6. Compare with your group members the types of symmetry you each found. Did any
show both rotational symmetry and line symmetry?
Examine this grid.
What types of symmetry are shown?
Conclusion:When a shape and its transformation image are drawn, the resulting diagram may show:
____ symmetry
__________ symmetry (_________________)
__________________ symmetry
Both _________ symmetry and ________________ symmetry
Ex. 1 For each pair of rectangle ABCD and EFGH, determine whether they are related by symmetry, and if so, what kind.
a.
b.
c
Ex. 2 Draw the image of rectangle ABCD after each transformation. Write the coordinates of each vertex and its image. Identify and describe the type of symmetry that results.
a. A rotation of 180 about the origin. ̊
b. A reflection in the x-axis.
c. A translation 4 units right and 1 unit down.
Note, that when writing translations, there is a shorthand form. For example, when writing the translation 4 units right and 1 unit down, we can write this as ____, ____.
Ex. 3 Write the shorthand form for the translation 7 units left and 2 units up.
Ex. 4 Draw the image of the pentagon PQRST after each translation below. Label the vertices of the pentagon’s image and list their coordinates. If each diagram has symmetry, describe it.
a. A translation L2.
b. A translation L2, D3.
Assignment Do #1 – 15 (omit #14) p. 373
Do Unit Review #1 – 19 p. 377
The unit test will be on _____________________________!
