Lesson 18 Understanding Reflections 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G., 8.G.3 1 Getting the idea A reflection is a tpe of transformation in which ou flip a figure across a line called the line of reflection. The starting figure is called the pre-image, and the figure after the reflection is called the image. The names of the points in the images of a transformation are shown using a prime smbol (9). A D image B B C C line of reflection A D pre-image The image formed b a reflection is congruent to the pre-image because the figure s size and shape do not change. Unlike other tpes of transformations, a reflection forms a mirror image, so the figures are congruent and the same distance from the line of reflection. Eample 1 Which figure shows the reflected image of a triangle? Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Strateg Figure C Figure A J L R J R K T S T S L K Figure B Figure D L B N L B N M D M D C Compare each image to the pre-image. Look for a mirror image. C Lesson 18: Understanding Reflections 189
Step 1 Compare each image to the pre-image. In Figure A, the image is formed b sliding nrst to a different location and b reducing its size. The image is not congruent to the pre-image. In Figure B, a congruent image is formed b turning nlmn around a point. In Figure C, a congruent image is formed b flipping n JKL over a line. In Figure D, the image is formed b sliding nbcd to a different location. The image is congruent to the pre-image. Step Identif which of the four figures shows a reflection of the pre-image. Figure C shows a reflection because the pre-image flips across a line of reflection to form a congruent mirror image. Solution The image in Figure C is a reflection of n JKL. Eample Verif that quadrilateral K9L9M9N9 is a reflection of quadrilateral KLMN b showing that (a) the sides of the image and the pre-image are congruent, (b) the angles of the image and the pre-image are congruent, and (c) corresponding vertices are the same distance from the line of reflection. N K K N M M L L Strateg Use a ruler to measure side lengths and the distance of points from the line of reflection. Use a protractor to measure angles. Step 1 Use a ruler to show that corresponding sides of the image and the pre-image are congruent. KL 5.9 cm and K9L9 5.9 cm, so KL > K9L9 LM 5.7 cm and L9M9 5.7 cm, so LM > L9M9 MN 5. cm and M9N9 5. cm, so MN > M9N9 NK 5. cm and N9K9 5. cm, so NK > N9K9 Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 190 Domain : Geometr
Step Use a protractor to show that corresponding angles of the image and the pre-image are congruent. m/k 5 m/k9 5 908, so /K > /K9 m/l 5 m/l9 5 88, so /L > /L9 m/m 5 m/m9 5 888, so /M > /M9 m/n 5 m/n9 5 1008, so /N > /N9 Step 3 Use a ruler to show that corresponding vertices of the image and the pre-image are the same distance from the line of reflection. Draw a line segment from each verte of quadrilateral KLMN to the corresponding verte of quadrilateral K9L9M9N9. Then measure the distance from each verte to the line of reflection. distance from point K to the line 5 distance from point K9 to the line 5 0.7 cm distance from point L to the line 5 distance from point L9 to the line 5 1. cm distance from point M to the line 5 distance from point M9 to the line 5 3.7 cm distance from point N to the line 5 distance from point N9 to the line 5.8 cm Since corresponding vertices of the image and the pre-image are the same distance from the line of reflection, the image and the pre-image are congruent. Solution Since corresponding sides are congruent, corresponding angles are congruent, and corresponding vertices are the same distance from the line of reflection, quadrilateral K9L9M9N9 is a reflection of quadrilateral KLMN. Eample 3 Show that quadrilateral EFGH and its image after a reflection, quadrilateral E9F9G9H9, are congruent. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Strateg E H G G F Use a ruler to measure side lengths and compare. Use a protractor to measure angles and compare. F H E Lesson 18: Understanding Reflections 191
Step 1 Use a ruler to show that corresponding sides of the image and the pre-image are congruent. EF 5 E9F9 5. cm, so EF > E9F9 FG 5 F9G9 5 1. cm, so FG > F9G9 GH 5 G9H9 5 1.7 cm, so GH > G9H9 HE 5 H9E9 5 1. cm, so HE > H9E9 Step Use a protractor to show that corresponding angles of the image and the pre-image are congruent. m/e 5 m/e9 5 908, so /E > /E9 m/f 5 m/f9 5 08, so /F > /F9 m/g 5 m/g9 5 108, so /G > /G9 m/h 5 m/h9 5 908, so /H > /H9 Step 3 Use a ruler to show that corresponding vertices of the image and the pre-image are the same distance from the line of reflection. distance from point E to the line 5 distance from point E9 to the line 5.8 cm distance from point F to the line 5 distance from point F9 to the line 5 0.5 cm distance from point G to the line 5 distance from point G9 to the line 5 0. cm distance from point H to the line 5 distance from point H9 to the line 5.1 cm Solution Quadrilateral EFGH and its image quadrilateral E9F9G9H9 are congruent because their corresponding sides and angles are congruent and corresponding vertices are the same distance from the line of reflection. Eample Reflect rectangle ABCD over the -ais. Label the image A9B9C9D9. Verif that pre-image ABCD and image A9B9C9D9 are congruent. Strateg 5 3 1 D A C B 5 3 1 0 1 3 5 1 3 5 Find the ordered pairs for the vertices of A9B9C9D9. Then use the ordered pairs to find side lengths and compare. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 19 Domain : Geometr
Step 1 Determine the coordinates of each verte on the pre-image. A(1, 1) B(3, 1) C(3, ) D(1, ) Step Reflect ABCD over the -ais, and record the coordinates of the image. C 5 D D C 3 1 B A A B 5 3 1 0 1 3 5 1 3 5 A9(1, 1) B9(3, 1) C9(3, ) D9(1, ) Notice that the -coordinates staed the same. Onl the sign on the -coordinates changed. Step 3 Find the lengths of each side and compare. AB 5 A9B9 5 CD 5 C9D9 5 units BC 5 B9C9 5 DA 5 D9A9 5 3 units Corresponding sides are congruent. Step Find the measure of each angle and compare. All of the angles in rectangles ABCD and A9B9C9D9 are right angles. Corresponding angles are congruent. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Solution The ordered pairs for the image A9B9C9D9 are A9(1, 1), B9(3, 1), C9(3, ), and D9(1, ). Corresponding sides and angles are congruent, so the pre-image and image are congruent. Lesson 18: Understanding Reflections 193
COACHED EXAMPLE Reflect rectangle JKLM over the -ais. Then verif that the image is congruent to the pre-image. J 5 K M L 3 1 5 3 1 0 1 3 5 Write the coordinates of each verte of the pre-image. J(, ) K(, ) L(, ) M(, ) Reflect rectangle JKLM over the -ais, and record the coordinates of each verte of the image. J9(, ) K9(, ) L9(, ) M9(, ) Find the lengths of each side, and compare. JK 5 5 units KL 5 5 units LM 5 5 units M J 5 5 units Corresponding sides are. Compare the measures of corresponding angles. All angles are angles, so corresponding angles are. The pre-image and image are congruent because. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 19 Domain : Geometr
3 LESSON LESSON PRACTICE PRACTICE 1 The figure shows quadrilateral ABCD reflected over the dashed line to produce quadrilateral A9B9C9D9. Based on this reflection, complete each statement. A 55 135 B B A D 105 5 C C D m/d9 5 m/b9 5 AB > B9C9 > What are the coordinates of each verte of the image A9B9C9D9 if quadrilateral ABCD is reflected over the -ais or the -ais? Reflected over the -ais: Reflected over the -ais: A9: (, ) A9: (, ) B9: (, ) B9: (, ) A B D C 0 C9: (, ) C9: (, ) D9: (, ) D9: (, ) Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 3 Renee created the following four figures to use as designs on wallpaper. Select the design(s) that shows a reflection over the dashed line. Circle all that appl. A. C. B. D. Lesson 18: Understanding Reflections 195
Which statements correctl describe the image formed if trapezoid DEFG is reflected over the dashed line shown? Circle all that appl. D E 10 G 0 F A. D9E9 > F9G9 B. /F9 is a right angle. C. m/e9 5 108 D. E9F9 is parallel to EF. E. F9G9 is parallel to D9E9. 5 Which graph shows the reflection over the -ais of a triangle with vertices at coordinates (1, 1), (, ), and (, 1)? A. C. 0 0 B. 0 D. 0 Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 19 Domain : Geometr
Each street sign shown below has a vertical line drawn on it. Select a sign in which the part of the figure on the left of the line is reflected over the line to form the part of the figure on the right of the line. Circle all that appl. R R A. B. C. D. 7 The vertices of a quadrilateral on a coordinate plane are: A(3, 1), B(5, 1), C(5, 5), D(3, 5). Part A What are the coordinates of each verte of the image formed if the original quadrilateral is reflected over the -ais? You ma use the grid to help ou. A9(, ) B9(, ) C9(, ) D9(, ) Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC Part B 5 3 1 5 3 1 0 1 3 5 1 3 5 Verif that each side of the image is congruent to the corresponding side of the pre-image. Record the lengths of the corresponding sides. Lesson 18: Understanding Reflections 197
8 Kim sews designs for a quilt b reflecting an image in different was. The figures below show the pre-image and the line of reflection. Draw the image that Kim can produce b each reflection. 9 The figure shows two designs that Leila used in stained glass decorations. The shaded area shows the starting shape for each decoration. f a r s w e d c b t u Design A Design B Part A Which one of the two decorations could she make b reflecting the starting triangle five times? Part B Describe five reflections Leila could use to make this decoration. In our answer, use the letters on the figure to indicate the triangles on either side of the line of reflection that the artist could use for each part of the decoration. Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 198 Domain : Geometr
10 The figure shows quadrilateral KLMN on a coordinate grid. KL is parallel to NM. Quadrilateral KLMN is reflected over the -ais. Select True or False for each statement about the image formed b the reflection. L M K N 0 A. K9L9 will be parallel to N9M9. True False B. /K > /K9 True False C. Verte K will be one unit from the -ais. True False D. /M > /N9 True False E. L9M9 > LM True False Duplicating an part of this book is prohibited b law. 015 Triumph Learning, LLC 11 The figure shows n ABC. Part A Draw and label the image n A9B9C9 b reflecting n ABC over the -ais. Part B Draw and label the image n A0B0C0 b reflecting n A9B9C9 over the -ais. Part C 0 Compare the coordinates of the pre-image with the two images. Describe how the coordinates change when the pre-image is reflected over the -ais. How do the change when the image is reflected over the -ais? A C B Lesson 18: Understanding Reflections 199