# Geometry Notes Chapter 12. Name: Period:

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1 Geometry Notes Chapter 1 Name: Period:

2 Vocabulary Match each term on the left with a definition on the right. 1. image A. a mapping of a figure from its original position to a new position. preimage B. a ray that divides an angle into two congruent angles 3. transformation C. a shape that undergoes a transformation 4. vector D. a quantity that has both a size and a direction Ordered Pairs E. the shape that results from a transformation of a figure Graph each ordered pair. 5. (0, 4) 6. (-3, ) 7. (4, 3) 8. (3, -1) 9. (-1, -3) 10. (-, 0) Congruent Figures Can you conclude that the given triangles are congruent? If so, explain why. 11. PQS and PRS 1. DEG and FGE Identify Similar Figures Can you conclude that the given figures are similar? If so, explain why. 13. JKL and JMN 14. rectangle PQRS and rectangle UVWX Angles in Polygons 15. Find the measure of each interior angle of a regular octagon. 16. Find the sum of the interior angle measures of a convex pentagon. 17. Find the measure of each exterior angle of a regular hexagon. 18. Find the value of x in hexagon ABCDEF. Extending Transformational Geometry 81

3 LESSON 1-1 Reflections Lesson Objectives (p. 84): Vocabulary 1. Isometry (p. 84): Key Concepts. Reflections (p. 85): 3. Reflections in the Coordinate Plane (p. 86): ACROSS THE x-axis ACROSS THE y-axis ACROSS THE y x 4. Get Organized Complete the graphic organizer. (p. 86). LINE OF REFLECTION IMAGE OF (a, b) EXAMPLE x-axis y-axis y x 6 Geometry

4 Name Date Class LESSON 1-1 Reading Strategies Use a Concept Map Use the concept map below to help you understand reflections. Definition A reflection is a transformation that moves a figure, called the preimage, by flipping it across a line. The reflected figure is called the image. Examples Reflections Reflections in the Coordinate Plane Reflections can be made across the x-axis (x, y) (x, y) across the y-axis (x, y) (x, y) across the line y x (x, y) (y, x) Tell whether each transformation appears to be a reflection Give the vertices of the image after it is reflected across the given line. 4. A(, 1), B(6, 1), C(4, 3) across the x-axis 5. N(1, ), P(3, 5), Q(3, 7), R(1, 6) across the line y x Draw the reflection of each figure across the line Holt Geometry

5 Name Date Class LESSON 1-1 Review for Mastery Reflections An isometry is a transformation that does not change the shape or size of a figure. Reflections, translations, and rotations are all isometries. A reflection is a transformation that flips a figure across a line. Reflection Not a Reflection The line of reflection is the perpendicular bisector of each segment joining each point and its image. A A C B B C Tell whether each transformation appears to be a reflection. 1.. Copy each figure and the line of reflection. Draw the reflection of the figure across the line Holt Geometry

6 Name Date Class LESSON 1-1 Review for Mastery Reflections continued Reflections in the Coordinate Plane Across the x-axis Across the y -axis Across the line y x y R(x, y) R(x, y) R(x, y) y y R(x, y) 0 x 0 x 0 R(y, x) x R(x, y) (x, y) (x, y) (x, y) (x, y) y x (x, y) (y, x) Reflect FGH with vertices F(1, 4), G(, 4), and H(4, 1) across the x-axis. The reflection of (x, y) is (x, y). F(1, 4) F(1, 4) G(, 4) G(, 4) H(4, 1) H(4, 1) Graph the preimage and image. F F y 0 G G H x H Reflect the figure with the given vertices across the line. 5. M(, 4), N(4, ), P(3, ); y -axis 6. T(4, 1), U(3, 4), V(, 3), W(0, 1); x -axis y y 0 x 0 x 7. Q(3, 1), R(, 4), S(, 1); x-axis 8. A(, 4), B(1, 1), C(5, 1); y x y y 0 x 0 x 7 Holt Geometry

7 Name Date Class LESSON 1-1 Practice B Reflections Tell whether each transformation appears to be a reflection Draw the reflection of each figure across the line Sam is about to dive into a still pool, but some sunlight is reflected off the surface of the water into his eyes. On the figure, plot the exact point on the water s surface where the sunlight is reflected at Sam. Reflect the figure with the given vertices across the given line. 8. A(4, 4), B(3, 1), C(1, ); y-axis 9. D(4, 1), E(, 3), F(1, 1); y x 10. P(1, 3), Q(, 3), R(, 1), S(1, 0); 11. J(3, 4), K(1, 1), L(1, 1), x-axis M(, 4); y x 4 Holt Geometry

8 LESSON 1- Translations Lesson Objectives (p. 831): Key Concepts 1. Translations (p. 83):. Translations in the Coordinate Plane (p. 83): HORIZONTAL VERTICAL GENERAL TRANSLATION TRANSLATION TRANSLATION ALONG VECTOR (a, 0) ALONG VECTOR (0, b) ALONG VECTOR (a, b) 63 Geometry

9 LESSON 1- CONTINUED 3. Get Organized Complete the graphic organizer. (p. 833). Definition Translations Example Nonexample 64 Geometry

10 Name Date Class LESSON 1- Reading Strategies Use a Concept Map Use the concept map below to help you understand translations. Definition A translation is a transformation in which all the points of a figure are moved the same distance in the same direction. Examples Translations Translations in the Coordinate Plane Translations can be horizontal along the vector a, 0 (x, y) (x a, y) vertical along the vector 0, b (x, y) (x, y b) in any direction along the vector a, b (x, y) (x a, y b) Tell whether each transformation appears to be a translation Give the vertices of the image after it is translated along the given vector. 4. A(, 5), B(1, 3), C(1, 5) along 3, 0 5. N(, ), P(1, 4), Q(7, 4), R(6, ) along 4, 5 Draw the translation of each figure along the given vector Holt Geometry

11 Name Date Class LESSON 1- Practice B Translations Tell whether each transformation appears to be a translation Draw the translation of each figure along the given vector Translate the figure with the given vertices along the given vector. 7. A(1, 3), B(1, 1), C(4, 4); 0, 5 8. P(1, ), Q(0, 3), R(1, ), S(0, 1); 1, 0 9. L(3, ), M(1, 3), N(, );, D(, ), E(, 4), F(1, 4), G(, );, A builder is trying to level out some ground with a front-end loader. He picks up some excess dirt at (9, 16) and then maneuvers through the job site along the vectors 6, 0,, 5, and 8, 10 to get to the spot to unload the dirt. Find the coordinates of the unloading point. Find a single vector from the loading point to the unloading point. 1 Holt Geometry

12 LESSON 1-3 Rotations Lesson Objectives (p. 839): Key Concepts 1. Rotation (p. 840):. Rotations in the Coordinate Plane (p. 840): BY 90 ABOUT THE ORIGIN BY 180 ABOUT THE ORIGIN 65 Geometry

13 LESSON 1-3 CONTINUED 3. Get Organized Complete the graphic organizer. (p. 841). Definition REFLECTION TRANSLATION ROTATION Example 66 Geometry

14 Name Date Class LESSON 1-3 Practice B Rotations Tell whether each transformation appears to be a rotation Draw the rotation of each figure about point P by ma Rotate the figure with the given vertices about the origin using the given angle of rotation. 7. A(, 3), B(3, 4), C(0, 1); D(3, ), E(4, 1), F(, ), G(1, 1); J(, 3), K(3, 3), L(1, ); P(0, 4), Q(0, 1), R(, ), S(, 3); The steering wheel on Becky s car has a 15-inch diameter, and its center is at (0, 0). Point X at the top of the wheel has coordinates (0, 7.5). To turn left off her street, Becky must rotate the steering wheel by 300. Find the coordinates of X when the steering wheel is rotated. Round to the nearest tenth. (Hint: How many degrees short of a full rotation is 300?) 0 Holt Geometry

15 LESSON 1-4 Compositions of Transformations Lesson Objectives (p. 848): Vocabulary 1. Composition of transformations (p. 848):. Glide reflection (p. 848): Key Concepts 3. Theorem (p. 848): Theorem Theorem 1-4- (p. 849): 67 Geometry

16 LESSON 1-4 CONTINUED 5. Theorem (p. 850): 6. Get Organized In each box, describe an equivalent transformation and sketch an example. (p. 850). Composition of Two Reflections Across parallel lines Across intersecting lines 68 Geometry

17 Name Date Class LESSON 1-4 Review for Mastery Compositions of Transformations A composition of transformations is one transformation followed by another. A glide reflection is the composition of a translation and a reflection across a line parallel to the vector of the translation. Reflect ABC across line along u v and then translate it parallel to u v. A B v Reflect ABC across line. C A B C Translate the image along u v. Draw the result of each composition of transformations. 1. Translate HJK along u v and then reflect. Reflect DEF across line k and it across line m. then translate it along u. m k v H J K D E u F 3. ABC has vertices A(0, 1), B(3, 4), and 4. QRS has vertices Q(, 1), R(4, ), C(3, 1). Rotate ABC 180 about the origin and S(1, 3). Reflect QRS across the and then reflect it across the x-axis. y-axis and then translate it along the vector 1, 3. y y 0 x 0 x 30 Holt Geometry

18 Name Date Class LESSON 1-4 Review for Mastery Compositions of Transformations continued Any translation or rotation is equivalent to a composition of two reflections. Composition of Two Reflections To draw two parallel lines of reflection that produce a translation: Draw PP _, a segment connecting a preimage _ point P and its corresponding image point P. Draw the midpoint M of PP. Draw the perpendicular bisectors of _ PM and _ PM. To draw two intersecting lines that produce a rotation with center C: Draw PCP, _ where P is a preimage point and P is its corresponding image point. Draw CX, the angle bisector of PCP. Draw the angle bisectors of PCX and PCX. Copy ABC and draw two lines of reflection that A produce the translation ABC ABC. Step 1 Draw CC and the midpoint M of CC. A B A B B C A B C C Step Draw the perpendicular bisectors of M C CM and CM. A B A B C M C Copy each figure and draw two lines of reflection that produce an equivalent transformation. 5. translation: 6. rotation with center C: JKL JKL PQR PQR L J L J Q R Q P P K K R 31 Holt Geometry

19 Name Date Class LESSON 1-4 Practice B Compositions of Transformations Draw the result of each composition of isometries. 1. Rotate XYZ 90 about point P. Reflect LMN across line q and then and then translate it along u v. translate it along u. 3. ABCD has vertices A(3, 1), B(1, 1), 4. PQR has vertices P(1, 1), Q(4, 1), C(1, 1), and D(3, 1). Rotate and R(3, 1). Reflect PQR across the ABCD 180 about the origin and then x-axis and then reflect it across y x. translate it along the vector 1, Ray draws equilateral EFG. He draws two lines that make a 60 angle through the triangle s center. Ray wants to reflect EFG across 1 and then across. Describe what will be the same and what will be different about the image of EFG compared to EFG. Draw two lines of reflection that produce an equivalent transformation for each figure. 6. translation: STUV STUV 7. rotation with center P: STUV STUV 8 Holt Geometry

20 Quiz for Lessons 1-1 Through 1-4 SECTION 1A 1-1 Reflections Tell whether each transformation appears to be a reflection. 1.. Copy each figure and the line of reflection. Draw the reflection of the figure across the line Translations Tell whether each transformation appears to be a translation A landscape architect represents a flower bed by a polygon with vertices (1, 0), (4, 0), (4, ), and (1, ). She decides to move the flower bed to a new location by translating it along the vector -4, -3. Draw the flower bed in its final position. 1-3 Rotations Tell whether each transformation appears to be a rotation Rotate the figure with the given vertices about the origin using the given angle of rotation. 10. A (1, 0), B (4, 1), C (3, ) ; R (-, 0), S (-, 4), T (-3, 4), U (-3, 0) ; Compositions of Transformations 1. Draw the result of the following composition of transformations. Translate GHJK along v and then reflect it across line m. 13. ABC with vertices A (1, 0), B (1, 3), and C (, 3) is reflected across the y-axis, and then its image is reflected across the x-axis. Describe a single transformation that moves the triangle from its starting position to its final position. Ready to Go On? 855

21 LESSON 1-5 Symmetry Lesson Objectives (p. 856): Vocabulary 1. Symmetry (p. 856):. Line symmetry (p. 856): 3. Line of symmetry (p. 856): 4. Rotational symmetry (p. 857): Key Concepts 5. Line Symmetry (p. 856): 6. Rotational Symmetry (p. 857): 69 Geometry

22 LESSON 1-5 CONTINUED 7. Get Organized In each region, draw a figure with the given type of symmetry. (p. 858). Line Symmetry Both Rotational Symmetry No Symmetry 70 Geometry

23 Name Date Class LESSON 1-5 Practice B Symmetry Tell whether each figure has line symmetry. If so, draw all lines of symmetry Anna, Bob, and Otto write their names in capital letters. Draw all lines of symmetry for each whole name if possible. Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry This figure shows the Roman symbol for Earth. Draw all lines of symmetry. Give the angle and order of any rotational symmetry. Tell whether each figure has plane symmetry, symmetry about an axis, both, or neither Holt Geometry

24 LESSON 1-6 Tessellations Lesson Objectives (p. 863): Vocabulary 1. Translation symmetry (p. 863):. Frieze pattern (p. 863): 3. Glide reflection symmetry (p. 863): 4. Tessellation (p. 863): 5. Regular tessellation (p. 864): 6. Semiregular tessellation (p. 864): Key Concepts 7. Get Organized Complete the graphic organizer. (p. 866). Regular Tessellations Semiregular Tessellations How are they alike? How are they different? 71 Geometry

25 Name Date Class LESSON 1-6 Practice B Tessellations Tell whether each pattern has translation symmetry, glide reflection symmetry, or both Use the given figure to create a tessellation Classify each tessellation as regular, semiregular, or neither Determine whether the given regular polygon(s) can be used to form a tessellation. If so, draw the tessellation Holt Geometry

26 LESSON 1-7 Dilations Lesson Objectives (p. 87): Vocabulary 1. Center of dilation (p. 873):. Enlargement (p. 873): 3. Reduction (p. 873): Key Concepts 4. Dilations (p. 873): 5. Dilations in the Coordinate Plane (p. 874): 7 Geometry

27 LESSON 1-7 CONTINUED 6. Get Organized Select values for a center and radius. Then use the center and radius you wrote to fill in the circles. Write the corresponding equation and draw the corresponding graph (p. 801). Scale Factor k k > 1 0 < k < 1 1 < k < 0 k < 1 73 Geometry

28 Name Date Class LESSON 1-7 Practice B Dilations Tell whether each transformation appears to be a dilation Draw the dilation of each figure under the given scale factor with center of dilation P. 5. scale factor: 1 6. scale factor: 7. A sign painter creates a rectangular sign for Mom s Diner on his computer desktop. The desktop version is 1 inches by 4 inches. The actual sign will be 15 feet by 5 feet. If the capital M in Mom s will be 4 feet tall, find the height of the M on his desktop version. Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 8. A(, ), B(, 3), C(3, 3), D(3, ); 9. P(4, 4), Q(3, 1), R(, 3); scale factor: 1 scale factor: J(0, ), K(, 1), L(0, ), M(,1); 11. D(0, 0), E(1, 0), F(1, 1); scale factor: scale factor: 5 Holt Geometry

29 Quiz for Lessons 1-5 Through 1-7 SECTION 1B 1-5 Symmetry Explain whether each figure has line symmetry. If so, copy the figure and draw all lines of symmetry Explain whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry Tessellations Copy the given figure and use it to create a tessellation Classify each tessellation as regular, semiregular, or neither Determine whether it is possible to tessellate a plane with regular octagons. If so, draw the tessellation. If not, explain why. 1-7 Dilations Tell whether each transformation appears to be a dilation Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin. 17. A (0, ), B (-1, 0), C (0, -1), D (1, 0) ; scale factor: 18. P (-4, -), Q (0, -), R (0, 0), S (-4, 0) ; scale factor: - 1 _ Ready to Go On? 881

30 Tell whether each transformation appears to be a reflection. 1.. Tell whether each transformation appears to be a translation An interior designer is using a coordinate grid to place furniture in a room. The position of a sofa is represented by a rectangle with vertices (1, 3), (, ), (5, 5), and (4, 6). He decides to move the sofa by translating it along the vector -1, -1. Draw the sofa in its final position. Tell whether each transformation appears to be a rotation Rotate rectangle DEFG with vertices D (1, -1), E (4, -1), F (4, -3), and G (1, -3) about the origin by Rectangle ABCD with vertices A (3, -1), B (3, -), C (1, -), and D (1, -1) is reflected across the y-axis, and then its image is reflected across the x-axis. Describe a single transformation that moves the rectangle from its starting position to its final position. 10. Tell whether the no entry sign has line symmetry. If so, copy the sign and draw all lines of symmetry. 11. Tell whether the no entry sign has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry. Copy the given figure and use it to create a tessellation Classify the tessellation shown as regular, semiregular, or neither. Tell whether each transformation appears to be a dilation Draw the image of ABC with vertices A (, -1), B (1, -4), and C (4, -4) under a dilation centered at the origin with scale factor Chapter 1 Extending Transformational Geometry

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