lesson 2 problem solving practice reflections answer key

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Math Connects: Concepts, Skills, and Problem Solving Course 3, Grade: 8 Publisher: Glencoe/McGraw-Hill

Math connects: concepts, skills, and problem solving course 3, title : math connects: concepts, skills, and problem solving course 3, publisher : glencoe/mcgraw-hill, isbn : 78740509, isbn-13 : 9780078740503, use the table below to find videos, mobile apps, worksheets and lessons that supplement math connects: concepts, skills, and problem solving course 3., textbook resources.

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Ray Optics: Reflection and Mirrors

Calculator pad, version 2, reflection and mirrors: problem set.

A light ray approaches a mirror at an angle of incidence of 25°. What is the angle of reflection?

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A light ray approaches a mirror at an angle of 22° with the mirror surface. What is the angle of reflection of this light ray?

Angle B = 38° Angle C = 52° Angle D = 38°

Anna Litical is doing the Plane Mirror Lab in physics class. She places a pin a distance of 4.9 cm from a plane mirror. How far behind the mirror can the image be expected to appear?

Baldwin Young stands 68 cm from his dresser mirror, inspecting his scalp. How far is the image of his scalp located from his scalp?

A meter stick (object) is placed in an upright position in front of a plane mirror as shown in the diagram at the right. The image of the meter stick is equidistant from the mirror. Suppose that the meter stick is equipped with a working eyeball capable of viewing the top and the bottom of its image. The eyeball is located at the 90-cm mark on the meter stick. Using either a ray diagram or geometry, determine … a. … the location of the intersection of the eye's line of sight with the mirror as the eyeball sights at the top of the image. b. … the location of the intersection of the eye's line of sight with the mirror as the eyeball sights at the bottom of the image. c. … the amount of mirror required by the meter stick to view the image.

a. 95 cm b. 45 cm c. 50 cm

A spherical concave mirror has a radius of curvature of +62 cm. What is the focal length of the mirror?

A decorative garden sphere has a diameter of 44 cm. The reflecting surface of the shiny sphere makes a great convex mirror. What is the focal point of the convex surface?

In a physics demonstration, a concave mirror having a 50.0 cm focal length is used to create images of a candle located at various locations along its principal axis. Beginning from a distance of several meters from the mirror, a candle is moved forward and its image is projected onto an opaque screen. Determine the image distances (distance from mirror to image) for object distances (distance from object to mirror) of … a. … 125.0 cm b. … 100.0 cm c. … 75.0 cm d. … 50.0 cm (Be careful with your math; the result is surprising.) e. … 25.0 cm

a. 83.3 cm b. 100.0 cm c. 150.0 cm d. No image. A solution to the mirror equation does not exist for this object distance. e. -50.0 cm

Problem 10:

Obtaining a large spherical mirror with a focal length of 0.654 m from the Physics Storeroom, Mr. H takes his last period class outside for a fascinating demo. A student volunteer holds the mirror at an angle such that the face of the mirror is directed towards the Sun - roughly 1.46x10 11 m away. Mr. H then uses a piece of paper with George Washington's picture on it to focus the image of the sun on the sheet of paper. Before the paper engulfs in flames, a bright image of the sun can be seen on the paper. Use the mirror equation to calculate the distance from the mirror to the image of the sun.

Problem 11:

Every morning Bob Gillette uses a shaving mirror with a focal length of 72 cm to view the image of his face. Supposing his face is 18 cm from the mirror, determine the image distance and the magnification of his face.

d i = -24 cm Magnification = 1.33

Problem 12:

The infamous Chinese magician Foo Ling Yu places a 56-mm tall light bulb a distance of 124 cm from a spherical concave mirror with a focal length of 62 cm. a. Determine the image distance and image height. b. Describe the orientation and type of the image.

a. d i = 124 cm and h i = -56 mm b. The image is inverted and real.

Problem 13:

In a physics lab, Anna Litical and Noah Formula position a small night light bulb at several locations along the principal axis of a concave mirror. Using a note card, they locate the image of the light bulb. The mirror has a focal length of 32.0 cm. What image distances would you expect Anna and Noah to observe when the object is located at distances of … a. … 85.3 cm from the mirror? b. … 64.0 cm from the mirror? c. … 48.1 cm from the mirror?

a. 51.2 cm b. 64.0 cm c. 95.6 cm

Problem 14:

Ima Primpin uses a cosmetic mirror to magnify her eyelashes during the traditional morning painting session. Her 1.2-cm long eyelashes are magnified to 1.6 cm when placed 5.8 cm from the mirror. a. Determine the image distance for such an upright image. b. Determine the focal length of the mirror.

a. -7.7 cm b. 23.2 cm

Problem 15:

In the Fall of 2006, the Sky Mirror sculpture was opened in Rockefeller Center in New York City. Standing three stories tall and weighing 23 tons, its concave side faced the Rockefeller Center and its convex side faced Fifth Avenue. a. A taxi on Fifth Avenue is located 38 m from the convex side of the sculpture and its image is one-fifth the size of the taxi. Determine the focal length of the mirror. b. Estimate the image size and image distance of the 260-m tall Rockefeller Center if it is located an estimated distance of 95 meters from the concave mirror surface. Assume the focal length of the concave side is the same magnitude as the focal length of the convex side.

a. -9.5 m b. d i = 11 m (rounded from 10.55 m) and h i = -29 m (- indicates inverted image)

Problem 16:

A convex spherical mirror has a focal point located a distance of 24.6 cm from the surface of the mirror. (You will have to decide for yourself whether f is + or -.) a. Find the image distance (in cm) for an object distance of 76.8 cm. b. Determine the magnification of this image.

a. d i = -18.6 cm b. M = 0.243

Problem 17:

A convenient store mounts a convex mirror in the corner of the store to serve as a security mirror and reduce the frequency of five-finger discounts . When Robin Storz is positioned a distance of 4.8 m from the mirror, her image is magnified by a factor of one-half. Determine the focal length of the mirror.

Problem 18:

Kerry Uss is studying the convex side of her soup spoon. She notices that her 3.8-cm tall nose appears to be 1.2 cm tall when positioned a distance of 2.4 cm from the spoon. a. Determine the image distance for this particular object distance. b. Determine the focal length of the convex side of the spoon.

a. d i = -0.76 cm b. f = -1.1 cm

Problem 19:

A large spherical mirror sculpture is constructed in the town square at Physicston, Illinois. The sculpture consists of a large sphere with a diameter of 24 meters which is coated with a reflecting material. A 1.8 meter tall photographer stands a distance of 38 m from the concave side of the sculpture and takes a picture. Determine the image distance and the magnification of the photographer.

d i = 7.1 m Mag = -0.19

Problem 20:

Baxter Nachure lives in the country along Sinewave road. It is difficult to pull out of the driveway onto the road since the road is curved and trees prevent him from seeing around the corner. He recently installed a large convex mirror at one of the curves to give him a wider angle of view. It has a focal length of -1.54 meters. Determine the magnification of an oncoming car located 35.8 m from the mirror.

Problem 21:

A virtual image is formed 26.9 cm from a concave mirror having a radius of curvature of 48.1 cm. Determine the object distance.

Problem 22:

An 4.9-cm tall object is positioned 14.8 cm from a mirror. Determine the radius of curvature which the mirror must have in order to produce an upright image that is 7.2 cm tall?

Problem 23:

A dentist uses a spherical mirror to produce an upright image of a patient's tooth which is magnified by a factor of 4.5 when placed 1.8 cm from the tooth. (a) What type of mirror - concave or convex - is being used? (b) What is the focal length of the mirror?

a. Concave mirror. Convex mirrors do not magnify the image; they only reduce the image. b. 2.3 cm

Problem 24:

The real image produced by a concave mirror is observed to be six times larger than the object when the object is 34.2 cm in front of a mirror. Determine the radius of curvature of this mirror.

Problem 25:

A shiny bauble (ornament) hangs on Mr. H's Christmas tree. The bauble has a radius of 4.8 cm. Matthew looks into the bauble and observes an image of his face which is one-eighth the size of his face. How far from the bauble is Matthew's face?

Problem 26:

A child at an amusement park stands in front of a concave mirror with a focal length of 73.9 cm. With great amusement, the child holds her cotton candy close to the mirror and observes that its image is magnified by a factor of five. Determine the object distance which creates this magnification of five.

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Go Math Grade 8 Answer Key Chapter 9 Transformations and Congruence

There are multiple ways to learn maths. But choosing the best material is also important for the students to score good marks and also to improve their knowledge. This will be possible only with the help of our Go Math Grade 8 Answer Key Chapter 9 Transformations and Congruence. You can get the explanation for all the questions in detail in Go Math Grade 8 Chapter 9 Transformations and Congruence Answer Key here. Refer to Go Math Grade 8 Answer Key for learning the problems in an easy manner.

Practice makes you perfect become a master in maths. Without proper knowledge on the subject, you cannot solve the problems in real-time. This will affect your exam. So, keeping this in our mind we have prepared the solutions briefly which will help you to understand the concept in depth. Get free access to Download Go Math Grade 8 Chapter 9 Transformations and Congruence Solution Key pdf. In this article, you will get the solutions according to the topics. Therefore, students who want to score good marks in the exam must practice with Go Math Grade 8 Answer Key Chapter 9 Transformations and Congruence.

Lesson 1: Properties of Translations

• Properties of Translations – Page No. 282
• Properties of Translations – Page No. 283
• Properties of Translations Lesson Check – Page No. 284

Lesson 2: Properties of Reflections

• Properties of Reflections – Page No. 288
• Properties of Reflections – Page No. 289
• Properties of Reflections Lesson Check – Page No. 290
• Properties of Reflections Lesson Check 1 – Page No. 294
• Properties of Reflections Lesson Check 2 – Page No. 295
• Properties of Reflections Lesson Check 3 – Page No. 296

Lesson 3: Algebraic Representations of Transformations

• Algebraic Representations of Transformations – Page No. 300
• Algebraic Representations of Transformations – Page No. 301
• Algebraic Representations of Transformations Lesson Check – Page No. 302

Lesson 4: Congruent Figures

• Congruent Figures – Page No. 306
• Congruent Figures – Page No. 307
• Congruent Figures Lesson Check – Page No. 308
• Model Quiz – Page No. 309

Mixed Review

• Mixed Review – Page No. 310

Guided Practice – Properties of Translations – Page No. 282

Question 1. Vocabulary A __________________is a change in the position, size, or shape of a figure. ____________

Answer: transformation

Explanation: A transformation is a change in the position, size, or shape of a figure.

Question 2. Vocabulary When you perform a transformation of a figure on the coordinate plane, the input of the transformation is called the ________________, and the output of the transformation is called the_________________ . Type below: ____________

Answer: pre-image image

Explanation: When you perform a transformation of a figure on the coordinate plane, the input of the transformation is called the pre-image, and the output of the transformation is called the image.

Question 3. Joni translates a right triangle 2 units down and 4 units to the right. How does the orientation of the image of the triangle compare with the orientation of the preimage? Orientation is: _______

Answer: Orientation is: Same

Explanation: Since translation does not change the shape and size of a geometric figure, the two triangles are identical in shape and size, so they are congruent and the orientation is the same

Question 4. Rashid drew rectangle PQRS on a coordinate plane. He then translated the rectangle 3 units up and 3 units to the left and labeled the image P ‘Q ‘R ‘S ‘. How do rectangle PQRS and rectangle P ‘Q ‘R ‘S ‘ compare? They are: _______

Answer: congruent

Explanation: Since translation does not change the shape and size of a geometric figure, the two rectangles are identical in shape and size, so they are congruent.

Answer: After translation: W'(-4, 3) X'(2, 3) Y'(1, 1) Z'(-3, 1)

ESSENTIAL QUESTION CHECK-IN

Question 6. What are the properties of translations? Type below: ____________

Answer: -> a translation is a geometric transformation that moves every point of a figure or space by the same amount in a given direction. -> So the figures are identical and are congruent.

9.1 Independent Practice – Properties of Translations – Page No. 283

Answer: 2 left, and 4 down

Question 7. b. How would you describe the translation? Type below: ____________

Answer: It has the same size, shape. and orientation, but a different location

Question 7. c. How does the image of triangle DEF compare with the preimage? ____________

Question 8. b. On the same coordinate grid, graph the image of quadrilateral KLMN after a translation of 3 units to the right and 4 units up. Type below: ____________

Question 8. c. Which side of the image is congruent to side \(\overline { LM } \)? ___________ Name three other pairs of congruent sides. ___________ Type below: ____________

Answer: Line LM is congruent to Line L!M! Line KL is equal to K’L’ Line MN is equal to M’N’ Line KN is equal to K’N’

Draw the image of the figure after each translation.

Answer: After translation P'(-3, 1) Q'(0, 2) R'(0, -1) S'(-3, -3)

Answer: After translation A'(0, 4) B'(3, 5) C'(3, 1) D'(0, 0)

Properties of Translations – Page No. 284

Answer: 4 units along positive X and 5 units along positive Y

Explanation: Initial coordinate of balloon = ( -2 , -4) Final coordinates of the balloon = (2,1) Translation along x axis = 2 – (-2) = 4 units along positive x direction Translation along y axis = 1-(-4) = 5 units along the positive y direction

Question 12. Critical Thinking Is it possible that the orientation of a figure could change after it is translated? Explain. _________

Answer: No, it is not possible to change the orientation just by translation. As translation means, a transformation in which a figure is moved to another location without any change in size or orientation.

FOCUS ON HIGHER ORDER THINKING

Question 13. b. On the same coordinate grid, graph and label triangle X’Y’Z’, the image of triangle XYZ after a translation of 3 units to the left and 6 units up.

Question 13. c. Now graph and label triangle X”Y”Z”, the image of triangle X’Y’Z’ after a translation of 1 unit to the left and 2 units down. Type below: ____________

Question 13. d. Analyze Relationships How would you describe the translation that maps triangle XYZ onto triangle X”Y”Z”? Type below: ____________

Answer: Triangle XYZ has translated 4 units up and 4 units to the left

Question 15. Communicate Mathematical Ideas Explain why the image of a figure after a translation is congruent to its preimage. Type below: ____________

Answer: A translation is a geometric transformation that moves every point of a figure or space by the same amount in a given direction. So the 2 figures are identical and the translated figure is congruent to its pre-image.

Guided Practice – Properties of Reflections – Page No. 288

Question 1. Vocabulary A reflection is a transformation that flips a figure across a line called the __________ . ____________

Answer: Reflection Axis

Explanation: A reflection is a transformation that flips a figure across a line called the Reflection Axis.

Answer: A'(-3, -4) B'(1, -4) C'(3, -1) D'(-3, -1)

Question 2. b. How do trapezoid ABCD and trapezoid A’B’C’D’ compare? ____________

Explanation: trapezoid ABCD and trapezoid A’B’C’D’ are congruent

Question 2. c. What If? Suppose you reflected trapezoid ABCD across the y-axis. How would the orientation of the image of the trapezoid compare with the orientation of the preimage? Type below: ____________

Answer: The orientation would be reversed horizontally.

Question 3. What are the properties of reflections? Type below: ____________

Answer: properties of reflections -> the size stays the same -> the shape stays the same -> the orientation does NOT stay the same

9.2 Independent Practice – Properties of Reflections – Page No. 289

Question 4. Which two triangles are reflections of each other across the x-axis? Type below: ____________

Answer: Triangles A and C are the reflections of each other across the x-axis.

Question 5. For which two triangles is the line of reflection the y-axis? Type below: ____________

Answer: For triangle C & D the line of reflection is y-axis.

Question 6. Which triangle is a translation of triangle C? How would you describe the translation? Type below: ____________

Answer: Triangle B is the translation of triangle C. Lets take any one point of the triangle = (-2, -6) Lets take the corresponding side of triangle B = (4,2) Translation across x axis = 4 -(-2) = 6 units Translation across y axis = 2 -(-6) = 8 units

Question 7. Which triangles are congruent? How do you know? Type below: ____________

Answer: All the 4 triangles A, B, C, D are congruent. The length of base and height of all the four triangles are 3 units, 4 units respectively.

Explanation: All the 4 triangles A, B, C, D are congruent. If base and height are equal then the hypotenuse should also be equal. Thus all three sides of the triangles A,B,C,D are equal. Thus these triangles are congruent, The length of base and height of all the four triangles are 3 units, 4 units respectively.

Question 8. a. Graph quadrilateral WXYZ with vertices W(-2, -2), X(3, 1), Y(5, -1), and Z(4, -6) on the coordinate grid. Type below: ____________

Question 8. b. On the same coordinate grid, graph quadrilateral W’X’Y’Z’, the image of quadrilateral WXYZ after a reflection across the x-axis. Type below: ____________

Question 8. c. Which side of the image is congruent to side \(\overline { YZ } \)? _______________ Name three other pairs of congruent sides. _______________ Type below: ____________

Answer: Line YZ = Line Y’Z’ Line WX = Line W’X’ Line XY = Line X’Y’ Line WZ = Line W’Z’

Question 8. d. Which angle of the image is congruent to ∠X? _______________ Name three other pairs of congruent angles. _______________ Type below: ____________

Answer: Angle X’ Angle W and Angle W’ Angle Y and Angle Y’ Angle Z and Angle Z’

Properties of Reflections – Page No. 290

Question 9. Critical Thinking Is it possible that the image of a point after a reflection could be the same point as the preimage? Explain. ________

Answer: Yes

Explanation: It is possible that the image of a point after a reflection could be the same point as the preimage

Question 10. b. On the same coordinate grid, graph the image of the figure you drew in part a after a reflection across the x-axis. Type below: ____________

Question 10. c. Make a Conjecture What other sequence of transformations would produce the same final image from the original preimage? Check your answer by performing the transformations. Then make a conjecture that generalizes your findings. Type below: ____________

Answer: The same image can be obtained by reflecting first across the x-axis and then across the y-axis. Reflecting a figure first across the y-axis and then across the x-axis has the same outcome,. reflecting first across the x-axis and then across the y-axis.

Question 11. a. Graph triangle DEF with vertices D(2, 6), E(5, 6), and F(5, 1) on the coordinate grid.

Question 11. b. Next graph triangle D ′E ′F ′, the image of triangle DEF after a reflection across the y-axis. Type below: ____________

Question 11. c. On the same coordinate grid, graph triangle D′′ E′′ F′′, the image of triangle D ′E ′F ′ after a translation of 7 units down and 2 units to the right. Type below: ____________

Question 11. d. Analyze Relationships Find a different sequence of transformations that will transform triangle DEF to triangle D ′′E ′′F ′′. Type below: ____________

Answer: Translate triangle DEF 7 units down and 2 units to the left. Then reflect the image across the y-axis.

Guided Practice – Properties of Reflections – Page No. 294

Question 1. Vocabulary A rotation is a transformation that turns a figure around a given _____ called the center of rotation. ____________

Answer: point

Explanation: A rotation is a transformation that turns a figure around a given point called the center of rotation.

Siobhan rotates a right triangle 90° counterclockwise about the origin.

Question 2. How does the orientation of the image of the triangle compare with the orientation of the preimage? Type below: ____________

Answer: Each leg in the preimage is perpendicular to its corresponding leg in the image.

Question 3. Is the image of the triangle congruent to the preimage? ______

Explanation: The image of the triangle is congruent to the preimage

Draw the image of the figure after the given rotation about the origin.

Answer: After 180° rotation A'(-2, -3) B'(-4, -1) C'(-2, 0) D'(0, -1)

Question 6. What are the properties of rotations? Type below: ____________

Answer: Rotations preserve size and shape but change orientation.

9.3 Independent Practice – Properties of Reflections – Page No. 295

Answer: ABC was rotated 90º counterclockwise about the origin

Question 7. b. What are the coordinates of the image? Type below: ____________

Answer: A'(3, 1) B'(2, 3) C'(-1, 4)

Answer: The figure was rotated 180º about the origin.

Question 8. b. Can you describe this as a transformation other than a rotation? Explain. ____________

Explanation: This can also be described as a reflection across the y-axis.

Answer: A 180º rotation about the origin will preserve the orientation of the H-shaped figure in the grid.

Question 10. A point with coordinates (-2, -3) is rotated 90° clockwise about the origin. What are the coordinates of its image? (_______ , _______)

Answer: (-3, 2)

Explanation: The new coordinates are (-3, 2)

Complete the table with rotations of 180° or 90°. Include the direction of rotation for rotations of 90°.

Properties of Reflections – Page No. 296

Answer: After 180° A'(4, 0) B'(2, -1) C'(0, 0) D'(2, 1)

Answer: After 270º counterclockwise rotation A'(1, 2) B'(2, -1) C'(4, 2)

Question 16. Is there a rotation for which the orientation of the image is always the same as that of the preimage? If so, what? ______

Explanation: A 360º rotation will always be the same as the original image

Answer: Figure A: 2 time(s) Figure B: 1 time(s) Figure C: 4 time(s)

Explanation: 2 times to return to original orientation 1 time to return to original orientation 4 times to return to original orientation

Question 18. Make a Conjecture Triangle ABC is reflected across the y-axis to form the image A′B′C′. Triangle A′B′C′ is then reflected across the x-axis to form the image A″B″C″. What type of rotation can be used to describe the relationship between triangle A″B″C″ and triangle ABC? Type below: ____________

Answer: Triangle A’B’C’ is a 90º rotation of triangle ABC Triangle A”B”C” is a 90º rotation of triangle A’B’C’ Therefore, Triangle A”B”C” is a 180º rotation of triangle ABC

Question 19. Communicate Mathematical Ideas Point A is on the y-axis. Describe all possible locations of image A′ for rotations of 90°, 180°, and 270°. Include the origin as a possible location for A. Type below: ____________

Answer: If Point A is on the y-axis, Point A’ will be on the x-axis for 190° and 270° rotations and on the y-axis for 180° rotation If point A is at the origin, A’ is at the origin for any rotation about the origin.

Guided Practice – Algebraic Representations of Transformations – Page No. 300

Answer: After a translation of 6 units to the right: X'(3, -2) Y'(5, 0) Z'(7, -6)

Question 2. Describe what happens to the x- and y-coordinates after a point is reflected across the x-axis. Type below: ____________

Answer: The x-coordinate remains the same, while the sign of the y-coordinate changes

Answer: The triangle is rotated 90º clockwise about the origin

Question 4. How do the x- and y-coordinates change when a figure is translated right a units and down b units? Type below: ____________

Answer: The x-coordinates increase by a, and the y-coordinates decrease by b

9.4 Independent Practice – Algebraic Representations of Transformations – Page No. 301

Write an algebraic rule to describe each transformation.Then describe the transformation.

Answer: algebraic rule (x, y) -> (x-2, y-5) transformation translation of 2 units to the left and 5 units down new coordinates M'(-4, -2) N'(-2, -2) O'(-1, -4) P'(-4, -4)

Answer: algebraic rule (x, y) -> (-x, -y) transformation rotation of 180º new coordinates A'(0, 0) B'(0, -3) C'(2, -3) D'(2, 0)

Question 7. Triangle XYZ has vertices X(6, -2.3), Y(7.5, 5), and Z(8, 4). When translated, X′ has coordinates (2.8, -1.3). Write a rule to describe this transformation. Then find the coordinates of Y′ and Z′. Type below: ____________

Answer: algebraic rule (x, y) -> (x-3.2, y+1) new coordinates Y'(4.3, 6) Z'(4.8, 5)

Question 8. Point L has coordinates (3, -5). The coordinates of point L′ after a reflection are (-3, -5). Without graphing, tell which axis point L was reflected across. Explain your answer. ____________

Answer: Point L was reflected on the y-axis. When you reflect a point across the y-axis, the sign of the x-coordinate changes, and the sign of the y-coordinate remains the same

Answer: The rectangle is translated 2 units to the left and 4 units down

Question 10. Parallelogram ABCD has vertices A(−2, −5\(\frac{1}{2}\)), B(−4, −5\(\frac{1}{2}\)),C(-3, -2), and D(-1, -2). Find the vertices of parallelogram A′B′C′D′ after a translation of 2 \(\frac{1}{2}\) units down. Type below: __________

Answer: after a translation of 2 \(\frac{1}{2}\) units A'(-2, -8) B'(-4, -8) C'(-3, -4 \(\frac{1}{2}\)) D'(-1, -4 \(\frac{1}{2}\))

Algebraic Representations of Transformations – Page No. 302

Answer: (x,y) –> (x+1,y-0.5) (x+1,y-0.5) –> (x+0.5,y-0.25)

Explanation: translation in units (x,y) –> (x+1,y-0.5) This step converts translation rule in units to translation rule in inches. (Divide by 2 since graph paper is half inch paper. (x+1,y-0.5) –> (x+0.5,y-0.25)

Question 12. Kite KLMN has vertices at K(1, 3), L(2, 4), M(3, 3), and N(2, 0). After the kite is rotated, K′ has coordinates (-3, 1). Describe the rotation, and include a rule in your description. Then find the coordinates of L′, M′, and N′. Type below: __________

Answer: rotation 90 counterclockwise rule (x, y) -> (-y, x) new coordinates L'(-4, 2) M'(-3, 3) N'(0, 2)

Answer: (-5, 5) has the same coordinates

Question 13. b. What is the equation of a line through the origin and this point? Type below: __________

Answer: x and y are equal so switching x and y has no effect on the coordinates

Question 13. c. Describe the transformation of the triangle. Type below: __________

Question 14. Critical Thinking Mitchell says the point (0, 0) does not change when reflected across the x- or y-axis or when rotated about the origin. Do you agree with Mitchell? Explain why or why not. _______

Answer: Yes, I agree with Mitchell

Explanation: Reflecting across the x-axis or y-axis changes the sign of the y or x coordinate 0 cannot change signs. Rotating about the origin does not change the origin (0, 0)

Question 15. Analyze Relationships Triangle ABC with vertices A(-2, -2), B(-3, 1), and C(1, 1) is translated by (x, y) → (x – 1, y + 3). Then the image, triangle A′B′C′, is translated by (x, y) → (x + 4, y – 1), resulting in A″B″C″. a. Find the coordinates for the vertices of triangle A″B″C″. Type below: __________

Answer: A”(-2-1+4, -2+3-1) = A”(1, 0) B”(-3-1+4, 1+3-1) = B”(0, 3) C”(1-1+4, 1+3-1) = C”(4, 3)

Question 15. b. Write a rule for one translation that maps triangle ABC to triangle A″B″C″. Type below: __________

Answer: (x, y) -> (x-1+4, y+3-1) (x, y) -> (x+3, y+2)

Guided Practice – Congruent Figures – Page No. 306

Answer: A. After transformation (1, 3) (1, 4) (4, 4) (4, 3) B. After transformation (3, -1) (4, -1) (4, -4) (3, -4) C. After transformation (1, -1) (2, -1) (2, -4) (1, -4) D. After transformation (1, 1) (1, 2) (4, 2) (4, 1) E. After transformation (-6, -1) (-6, 0) (-3, 0) (-3, -1)

Question 2. What transformation is used to transform figure A into figure B? Type below: __________

Answer: Reflection across the y-axis

Explanation: Reflection across the y-axis is used to transform figure A into figure B

Question 3. What transformation is used to transform figure B into figure C? Type below: __________

Answer: Translation 3 units right and 4 units down

Explanation: Translation 3 units right and 4 units down is used to transform figure B into figure C

Question 4. What sequence of transformations is used to transform figure A into figure C? Express the transformations algebraically. Type below: __________

Answer: Reflection across the y-axis is used to transform figure A into figure B Translation 3 units right and 4 units down is used to transform figure B into figure C Algebraically: (x, y) -> (-x, y) (x, y) -> (x +3, y-4)

Question 5. Vocabulary What does it mean for two figures to be congruent? Type below: __________

Answer: Two figures are congruent when the figures have the same size and the same shape.

Question 6. After a sequence of translations, reflections, and rotations, what is true about the first figure and the final figure? Type below: __________

Answer: After a sequence of translations, reflections, and rotations, the first and final figures have the same size and shape. (They are congruent)

9.5 Independent Practice – Congruent Figures – Page No. 307

For each given figure A, graph figures B and C using the given sequence of transformations. State whether figures A and C have the same or different orientation.

Answer: Different orientation

Congruent Figures – Page No. 308

Answer: From Site A to Site B: Translation 2 units right and 4 units down The size did NOT change The orientation changed

Answer: From figure A to D: Reflection across the x-axis (-1, -5) (-1, -6) (2, -5) (4, -6) 90º clockwise rotation (4, -1) (5, -1) (5, -4) (4, -6) translation 6 units left (4, -1) (5, -1) (5, -4) (4, -6)

Question 13. Counterexamples The Commutative Properties for Addition and Multiplication state that the order of two numbers being added or multiplied does not change the sum or product. Are translations and rotations commutative? If not, give a counterexample. ________

Answer: No, Translation and rotations are not commutative

Explanation: The point (2, 2) becomes (2, -4) when translated 2 units to the right then rotated 90 around the origin. The point (2, 2) becomes (4, -2) when rotated 90 around the origin then translated 2 units to the right. The above two points are not the same.

Question 14. Multiple Representations For each representation, describe a possible sequence of transformations. a. (x, y) → (-x – 2, y + 1) Type below: ____________

Answer: translation 2 units right and 1 unit up reflection across y-axis

Question 14. b. (x, y) → (y, -x – 3) Type below: ____________

Answer: rotation 90º clockwise around the origin translation 3 units down

Ready to Go On? – Model Quiz – Page No. 309

9.1–9.3 Properties of Translations, Reflections, and Rotations

Answer: After translation: A'(2, 1) B'(2, -1) C'(5, -1)

Question 2. On the same coordinate grid, graph the image of triangle ABC after a reflection across the x-axis. Label the vertices of the image A”, B”, and C”. Type below: ____________

Answer: After reflection: A”(-4, -5) B”(-4, -3) C”(-1, -3)

Answer: After rotation: H'(0, -4) I'(0, -1) J'(2, -2) K'(2, -3)

9.4 Algebraic Representations of Transformations

Question 4. A triangle has vertices at (2, 3), (−2, 2), and (−3, 5). What are the coordinates of the vertices of the image after the translation (x, y) → (x + 4, y − 3)? Type below: ____________

Answer: After translation: (6, 0) (2, -1) (1, 2)

9.5 Congruent Figures

Question 5. Vocabulary Translations, reflections, and rotations produce a figure that is _____ to the original figure. Type below: ____________

Explanation: Vocabulary Translations, reflections, and rotations produce a figure that is congruent to the original figure.

Question 6. Use the coordinate grid for Exercise 3. Reflect H’I’J’K’ over the y-axis, then rotate it 180° about the origin. Label the new figure H″I″J″K″. Type below: ____________

Answer: after reflection H'(0, -4) I'(0, -1) J'(-2, -2) K'(-2, -3) after rotation H”(0, 4) I”(0, 1) J”(2, 2) K”(2, 3)

ESSENTIAL QUESTION

Question 7. How can you use transformations to solve real-world problems? Type below: ____________

Answer: Transformational properties allow the systematic movement of congruent figures while maintaining or adjusting their orientation.

Selected Response – Mixed Review – Page No. 310

Answer: c. C

Explanation: After a translation of 8 units right and 3 units up, the orientation of figure L stays the same.

Question 2. Figure A is reflected over the y-axis and then lowered 6 units. Which sequence describes these transformations? Options: a. (x, y) -> (x, -y) and (x, y) -> (x, y – 6) b. (x, y) -> (-x, y) and (x, y) -> (x, y – 6) c. (x, y) -> (x, -y) and (x, y) -> (x – 6, y) d. (x, y) -> (-x, y) and (x, y) -> (x – 6, y)

Answer: b. (x, y) -> (-x, y) and (x, y) -> (x, y – 6)

Explanation: reflection over y-axis: (x, y) -> (-x, y) Translation 6 units down (x, y) -> (x, y-6)

Answer: d. IV

Explanation: After a rotation of 90° counterclockwise about the origin, the triangle will be in QIV

Question 4. Which rational number is greater than −3 \(\frac{1}{3}\) but less than −\(\frac{4}{5}\)? Options: a. −0.4 b. −\(\frac{9}{7}\) c. −0.19 d. −\(\frac{22}{5}\)

Answer: b. −\(\frac{9}{7}\)

Question 5. Which of the following is not true of a trapezoid that has been reflected across the x-axis? Options: a. The new trapezoid is the same size as the original trapezoid. b. The new trapezoid is the same shape as the original trapezoid. c. The new trapezoid is in the same orientation as the original trapezoid. d. The x-coordinates of the new trapezoid are the same as the x-coordinates of the original trapezoid.

Answer: d. The x-coordinates of the new trapezoid are the same as the x-coordinates of the original trapezoid.

Explanation: The x-coordinates of the new trapezoid are the same as the x-coordinates of the original trapezoid.

Question 6. A triangle with coordinates (6, 4), (2, −1), and (−3, 5) is translated 4 units left and rotated 180° about the origin. What are the coordinates of its image? Options: a. (2, 4), (-2, -1), (-7, 5) b. (4, 6), (-1, 2), (5, -3) c. (4, -2), (-1, 2), (5, 7) d. (-2, -4), (2, 1), (7, -5)

Answer: d. (-2, -4), (2, 1), (7, -5)

Question 7. A rectangle with vertices (3, -2), (3, -4), (7, -2), (7, -4) is reflected across the x-axis and then rotated 90° counterclockwise. a. In what quadrant does the image lie? ____________

Answer: After reflection and rotation, the image lies in QII

Question 7. b. What are the vertices of the image? Type below: ____________

Answer: image vertices (-2, 3) (-4, 3) (-2, 7) (-4, 7)

Question 7. c. What other transformations produce the same image? Type below: ____________

Answer: A reflection across the y-axis and 90º clockwise rotation will produce the same result.

Conclusion:

The Go Math Grade 8 Answer Key Chapter 9 Transformations and Congruence pdf are available both online and offline. We have provided in pdf format so that students can practice the problems offline. We know that maths is the scoring and also typical among all the subjects. But you can make it easy if you understand the concept of the chapter. Students can refer to the Go Math Grade 8 Answer Key in their convenient way. Practice well and make maths your favorite subject. Best of Luck!!!

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