Contents Section Congruent Triangles Flip, Turn, Resize, and Slide 1 Transformed Triangles 2 Constructing Parallel Lines 5 Transformations 6 Reflections 7 Rotations 10 Summary 13 Check Your Work 14 Additional Practice Answers to Check Your Work Student Activity Sheets Credits Illustrations 1 (top left and right) Encyclopædia Britannica, Inc. 2013 Encyclopædia Britannica, Inc.
Congruent Triangles Flip, Turn, Resize, and Slide Here you see a design Michael made using a drawing tool on his computer. File Edit View Help Then he revised his design as follows: File Edit View Help A A He did this by using one of the buttons you see below. Rotate Right 90 Rotate Left 90 Flip Vertical Flip Horizontal Flip up or down Flip to the left or to the right 1. Which of the buttons did he use? 2. Starting again with his original design, which of the buttons did he use to transform the figure below? Congruent Triangles 1
Transformed Triangles For this activity you can cut out the triangle of Student Activity Sheet 7 or use a computer program that allows you to draw, flip, and rotate a figure. Draw your results on Student Activity Sheet 7. Draw a right triangle that looks like the figure on the left. Flip the triangle horizontally to the right. On Student Activity Sheet 7 draw the result. Start with the drawing of the original triangle. Rotate the triangle 90⁰ to the right. On Student Activity Sheet 7 draw the result. Start with the drawing of the original triangle. Flip the triangle vertically up and then rotate 90⁰ to the left. On Student Activity Sheet 7 draw the result. 3. What combination of moves can you use to transform triangle A into triangle B? A Rotate Right 90 Rotate Left 90 B Flip Vertical Flip Horizontal Find another combination of moves to transform triangle A into triangle B. Use Student Activity Sheet 8 to show your work. 2 It s All the Same
Congruent Triangles 4. Alicia has drawn a rectangle on the computer. Her computer program has several options for resizing a figure. Alicia selects the Stretch option and enters 150% for both the horizontal and the vertical dimensions. Stretch Horizontal: 150 % Vertical: 150 % a. What effect will this have on her rectangle? Illustrate your answer with an example. For another rectangle Alicia enters 25% for both the horizontal and the vertical dimensions. Stretch Horizontal: 25 % Vertical: 25 % b. What effect will this have on her rectangle? Illustrate your answer with an example. Previously, you learned about a multiplication factor. c. What is the multiplication factor in a? And in b? 5. Suppose Alicia enters the following: Stretch Horizontal: 150 % Vertical: 50 % a. What will happen to her rectangle? b. Why is the resulting rectangle not similar to the original one? Use an example to explain. Congruent Triangles 3
Congruent Triangles A When using some drawing programs on a computer, you can slide a figure to another position. The arrows in the picture show how triangle A can be moved to a new position. 6. a. Copy the triangle and arrow below on graph paper. b. Slide the triangle as indicated by the arrow and draw its new position. In the drawing at the top of the page, all of the arrows have the same direction and the same length so they are parallel. Using graph paper makes it easier to draw parallel lines. 7. Reflect Why do you think it is easier to draw parallel arrows on graph paper? 4 It s All the Same
Constructing Parallel Lines For this activity, you will need some paper, a straightedge or ruler, and a triangle template or one cut out from stiff cardboard. In this activity you will learn how to construct families of parallel lines like a draftsperson or a designer. i. Use a straightedge to draw a straight line. ii. Place a plastic or cardboard triangle along the line that you drew. (A triangle of any shape will work.) iii. Place the straightedge against another side of the triangle. iv. While holding the straightedge still, slide the triangle along the straightedge. v. Draw a second line along the edge of your triangle that is parallel to the first line. vi. Repeat step iv and draw a third line, parallel to the first two lines. How do you know that these three lines are parallel? Congruent Triangles 5
Congruent Triangles 8. On Student Activity Sheet 8, copy the image below and slide the triangle as indicated by the arrow. Draw the new triangle. Make sure to draw the parallel lines from the vertices of the original triangle to the vertices of the new triangle as precisely as possible. Transformations In mathematics there are operations that are similar to those in the drawing software. In mathematics however, the names are different and the operations have more rules. These operations are called transformations. The operations and their drawing software counterparts are listed below: Reflection: flipping Rotation: turning Dilation: resizing Translation: sliding 9. a. For which transformation(s) is the resulting figure always congruent to the original figure? b. What can you tell about the transformation(s) that do not produce a congruent figure? 6 It s All the Same
Congruent Triangles Reflections A reflection is a figure that is flipped across a line. The resulting figure is a mirror image, or reflection. In the drawing below, the left figure is flipped across line l. Line l is called the line of reflection. 10. How can you check by folding the drawing that the reflection is drawn correctly? 11. On Student Activity Sheet 9, triangle A and its reflection, triangle B, are shown. Draw the line of reflection as accurately as possible. A B Congruent Triangles 7
Congruent Triangles In the drawing below, ABC is reflected across line l. Its mirror image is written as A B C. (A is read as A prime. ) C C A B B A l 12. a. When you connect a point to its mirror image with a line, what can you say about the angle created by this line and the line of reflection? b. Check whether this is also true for problem 11. 13. Use Student Activity Sheet 9 for the following problems. a. Reflect KLM across line l. Draw and label the mirror image K L M. b. Reflect PQR across line l. Draw and label the mirror image P Q R. M L K R Q l P 8 It s All the Same
Congruent Triangles 14. Copy the two triangles below. Use the dotted line as a line of reflection to create a reflection of each shape. What is the difference in the resulting images? If you can reflect a figure over a line and the figure appears unchanged, then the figure has reflectional symmetry and the line of reflection is called a line of symmetry. A figure that contains a line of reflection has line symmetry. Some symmetrical shapes have more than one line of symmetry, as shown in the figure below. Congruent Triangles 9
Congruent Triangles 15. Identify which shapes below have line symmetry. For those shapes, draw all the lines of symmetry on Student Activity Sheet 10. Rotations 16. Triangles can have lines of symmetry. Draw a triangle showing each characteristic below if possible. a. a single line of symmetry b. two lines of symmetry b. three lines of symmetry A rotation is a transformation that turns a figure around a point along a circular path. On the right you see ABC rotated 90 clockwise around point P to create A B C. 17. Use Activity Sheet 11 to rotate ABC 90 counterclockwise around point P to create A B C. (A is read as A double prime. ) 18. By how many degrees does a figure need to be rotated to bring it back to its original location? A P C A B B C 10 It s All the Same
Congruent Triangles 19. P M K L KLM is rotated around point P. a. Use Activity Sheet 11 to label the new image K L M. b. By how many degrees and in which direction was KLM rotated to create K L M? c. Jamal says there is another angle of rotation possible for problem b above. By how many degrees and in which direction did he rotate KLM to create K L M? 20. Copy the two figures below. Rotate each figure 180 around the point inside the figure. For each, explain how the resulting image and the original image are related. Congruent Triangles 11
Congruent Triangles A figure has rotational symmetry when it can be rotated less than 360 and it still appears the same. The point of rotation is inside a figure with rotational symmetry. For example, a square can be rotated 90, 180, and 270 and still have the same appearance. 21. Why is there a restriction that the rotation should be less than 360? 22. On Student Activity Sheet 12, label the shapes that have rotational symmetry. Identify the point of rotation and the angle(s) of rotation. 12 It s All the Same
Transformations are used to make similar or congruent images of the original figure. Two figures are similar if they have the same shape but not necessarily the same size. Corresponding angles of similar figures are equal. Two figures are congruent when they are exactly the same shape and size. Corresponding angles and sides of congruent figures are equal. Reflection Flip across a line Rotation Turn around a point Translation Slide to a different location Dilation Resize using a multiplication factor that is the same in all directions A figure that contains a line of reflection has line symmetry. A figure has rotational symmetry when it can be rotated less than 360 and still have the same appearance. Congruent Triangles 13
Congruent Triangles 1. On Student Activity Sheet 13, translate the triangle below as indicated by the arrow. 2. On Student Activity Sheet 13, rotate the triangle below 90 counterclockwise around point P. P 14 It s All the Same
3. a. On Student Activity Sheet 14, flip KLM across line l. Draw and label the mirror image K L M. b. Flip PQR across line l. Draw and label the mirror image P Q R. M l K L R Q P 4. a. For any figures below that have line symmetry, draw all lines of symmetry. b. For any figures below that have rotational symmetry, identify the point of rotation and label the angle(s) of rotation. You have worked with many classifications of triangles. Is there a specific triangle that has rotational symmetry? Explain your answer. Congruent Triangles 15
Additional Practice Congruent Triangles 1. Chuck traces this parallelogram to the left and cuts it out. a. Chuck says he can fold his parallelogram in half so that the two parts fit together exactly. Do you agree? Explain. b. Are there any types of parallelograms that can be folded together in half so that the two parts fit together exactly? Explain. 2. Parallelograms A and B are congruent. a. In your notebook, trace parallelograms A and B. Show how to fit parallelogram A directly on top of parallelogram B using one or more of these transformations: translation, rotation, and reflection. B A b. Is there more than one possible solution? Explain. c. Which transformation (translation, rotation, or reflection) is required for all possible solutions? 1 4 2 3 5 3. All the triangles in the picture to the left are copies of triangle 1. They were made using rotation, translation, reflection, and a combination of these moves. Describe how you could move triangle 1 to make copies 2, 3, 4, and 5. 1 2: 1 3: 1 4: 1 5: 16 It s All the Same
Congruent Triangles 1. 2. P Answers to Check Your Work 17
Answers to Check Your Work 3. M l M K K L R Q R Q L P P 4. a. The trapezoid, the rhombus, the square, and the rectangle have line symmetry. b. The non-rectangular parallelogram, the rhombus, the square, and the rectangle have rotational symmetry. 180 180 90, 180, 270 180 18 It s All the Same
Name Student Activity Sheet 7 Use with page 2. Activity Flipped Horizontal: Original triangle: Rotated right 90 : Flipped vertical and rotated left 90 : Encyclopædia Britannica, Inc. This page may be reproduced for classroom use. Student Activity Sheet 7 Congruent Triangles 19
Student Activity Sheet 8 Use with page 2. Name Problem 3 What combination of moves can you use to transform triangle A into triangle B? Find another combination of moves to transform triangle A into triangle B. A Rotate Right 90 Rotate Left 90 B Flip Vertical Flip Horizontal Problem 8 Slide the triangle as indicated by the arrow. Be sure you construct the parallel arrows precisely. 20 It s All the Same Student Activity Sheet 8 Encyclopædia Britannica, Inc. This page may be reproduced for classroom use.
Name Student Activity Sheet 9 Problem 11 Triangle A and its reflection triangle B are shown below. Draw the line of reflection as accurately as possible. Problem 13 a. Reflect KLM across line l. Draw and label the mirror image K L M. b. Reflect PQR across line l. Draw and label the mirror image P Q R. Encyclopædia Britannica, Inc. This page may be reproduced for classroom use. Student Activity Sheet 9 It s All the Same 21
Student Activity Sheet 10 Name Problem 15 Identify which shapes below have line symmetry and for those shapes identified, draw all the lines of symmetry. 22 It s All the Same Student Activity Sheet 10 Encyclopædia Britannica, Inc. This page may be reproduced for classroom use.
Name Student Activity Sheet 11 Problem 17 Rotate ABC 90 counterclockwise around point P to create A B C (A is read as A double prime). Encyclopædia Britannica, Inc. This page may be reproduced for classroom use. Problem 19: KLM is rotated around point P. a. Label the new image K L M. b. By how many degrees and in which direction was KLM rotated to create K L M? c. Jamal says there is another angle of rotation possible for question b above. By how many degrees and in which direction did he rotate KLM to create K L M? Student Activity Sheet 11 It s All the Same 23
Student Activity Sheet 12 Name Problem 22 Label the shapes that have rotational symmetry, and identify the point of rotation and the angle(s) of rotation. 24 It s All the Same Student Activity Sheet 12 Encyclopædia Britannica, Inc. This page may be reproduced for classroom use.
Name Student Activity Sheet 13 Check Your Work Problem 1 Translate the triangle below as indicated by the arrow. Problem 2 Rotate the triangle below 90 counterclockwise around point P. Encyclopædia Britannica, Inc. This page may be reproduced for classroom use. Student Activity Sheet 13 It s All the Same 25
Student Activity Sheet 14 Name Problem 3 a. Flip KLM across line l. Draw and label the mirror image K L M. b. Flip PQR across line l. Draw and label the mirror image P Q R. Problem 4 a. For any figures below that have line symmetry, draw all lines of symmetry. b. For any figures below that have rotational symmetry, identify the point of rotation, and label the angles of rotation. 26 It s All the Same Student Activity Sheet 14 Encyclopædia Britannica, Inc. This page may be reproduced for classroom use.