5.7 Reflections and Symmetry

Transcription

1 Page of Reflections and Setr oal Identif and use reflections and lines of setr. Ke Words iage p. 52 reflection line of setr reflection is a transforation that creates a irror iage. The original figure is reflected in a line that is called the line of reflection. PROPERTIES O RELETIONS The reflected iage is congruent to the original figure. 2 The orientation of the reflected iage is reversed. 3 The line of reflection is the perpendicular bisector of the segents joining the corresponding points. original r line of reflection iage EXMPLE Identif Reflections Tell whether the red triangle is the reflection of the blue triangle in line. Visualize It! clockwise orientation counterclockwise orientation heck to see if all three properties of a reflection are et. Is the iage congruent to the original figure? Yes. 2 Is the orientation of the iage reversed? Yes. T has a clockwise orientation. T has a counterclockwise orientation. 3 Is the perpendicular bisector of the segents connecting the corresponding points? Yes. To check, draw a diagra and connect the corresponding endpoints. NSWER ecause all three properties are et, the red triangle is the reflection of the blue triangle in line. 282 hapter 5 ongruent Triangles

2 Page 2 of 9 EXMPLE 2 Identif Reflections Tell whether the red triangle is the reflection of the blue triangle in line. heck to see if all three properties of a reflection are et. Is the iage congruent to the original figure? Yes. 2 Is the orientation of the iage reversed? No. NSWER The red triangle is not a reflection of the blue triangle. Student elp VOULRY TIP Use the following relationship to help ou reeber that a reflection is a flip: reflection flip EXMPLE a. Which segent is the reflection of &* in the -ais? Which point corresponds to? to? b. Which segent is the reflection of &* in the -ais? Which point corresponds to? to? 3 Reflections in a oordinate Plane (, 3) ( 4, ) J( 4, ) K(, 3) a. The -ais is the perpendicular bisector of J&* and K&*, so the reflection of &* in the -ais is JK&*. ( 4, ) J( 4, ) is reflected onto J. (, 3) K(, 3) is reflected onto K. b. The -ais is the perpendicular bisector of &* and E&*, so the reflection of &* in the -ais is E &*. ( 4, ) (4, ) is reflected onto. (, 3) E(, 3) is reflected onto E. E(, 3) (4, ) Identif Reflections Tell whether the red figure is a reflection of the blue figure. If the red figure is a reflection, nae the line of reflection Reflections and Setr 283

3 Page 3 of 9 Setr In the photo, the irror s edge creates a line of setr. figure in the plane has a line of setr if the figure can be reflected onto itself b a reflection in the line. line of setr is a line of reflection. Visualize It! You a want to draw a shape on paper, cut it out, and then fold it to find the lines of setr. EXMPLE eterine the nuber of lines of setr in a square. 4 eterine Lines of Setr Think about how an different was ou can fold a square so that the edges of the figure atch up perfectl. vertical fold horizontal fold diagonal fold diagonal fold NSWER square has four lines of setr. EXMPLE 5 eterine Lines of Setr eterine the nuber of lines of setr in each figure. a. b. c. a. 2 lines of setr b. no lines of setr c. 6 lines of setr 284 hapter 5 ongruent Triangles

4 Page 4 of 9 eepiece Kaleidoscopes EXMPLE 6 Use Lines of Setr Mirrors are used to create iages seen through a kaleidoscope. The angle between the irrors is a. irror casing black glass irror irror irror angle glass cover colored glass or liquid Top view Iage seen b viewer KLEIOSOPES The parts of a kaleidoscope are shown above. pplication Links L S S Z O N E. O M ind the angle easure used to create the kaleidoscope design. Use the equation a 8 0, where n is the nuber of lines of n setr in the pattern. a. b. c. a. The design has 3 lines of setr. So, in the forula, n 3. a n 3 b. The design has 4 lines of setr. So, in the forula, n 4. a n 4 c. The design has 6 lines of setr. So, in the forula, n 6. a n 6 eterine Lines of Setr eterine the nuber of lines of setr in the figure Reflections and Setr 285

5 Page 5 of Eercises uided Practice Vocabular heck. oplete the stateent: figure in the plane has a(n)? if the figure can be reflected onto itself b a(n)? in the line. Skill heck eterine whether the red figure is a reflection of the blue figure lowers eterine the nuber of lines of setr in the flower Practice and pplications Etra Practice See p Identifing Reflections eterine whether the figure in red is a reflection of the figure in blue. Eplain wh or wh not oework elp Eaple : Es. 8 0 Eaple 2: Es. 8 0 Eaple 3: Es. 6 Eaple 4: Es Eaple 5: Es Eaple 6: Es Reflections in a oordinate Plane Tell whether the grid shows a reflection in the -ais, the -ais, or neither E E 286 hapter 5 ongruent Triangles

6 Page 6 of 9 Student elp SKILLS REVIEW To review coordinates, see p Reflections in a oordinate Plane In Eercises 4 6, use the diagra at the right. 4. Which segent is the reflection of &* in the -ais? Which point corresponds to? to? 5. Which segent is the reflection of &* in the -ais? Which point corresponds to? to? 6. opare the coordinates for &* with the coordinates for its reflection in the -ais. ow are the coordinates alike? ow are the different? E Visualize It! Trace the figure and draw its reflection in line k k 9. k k 20. Paper olding ollow these steps. old a piece of paper in half, twice. 2 raw a triangle and cut it out. 3 Unfold the paper and label the sections. Which of the triangles are reflections of the triangle in section? Eplain. Setr ecide whether the line shown is a line of setr Lines of Setr eterine the nuber of lines of setr Reflections and Setr 287

7 Page 7 of 9 You be the Judge eterine whether all lines of setr are shown. If not, sketch the figure and draw all the lines of setr areers 30. Visualize It! piece of paper is folded in half and soe cuts are ade as shown. Sketch the figure that represents the piece of paper unfolded. Tpe esign In Eercises 3 and 32, use the lowercase letters of the alphabet shown below. 3. Which letters are reflections of other letters? TYPE ESINERS design fonts that appear in books, agazines, newspapers, and other aterials that we read ever da. Erik Spiekerann, shown above, has designed an fonts that are widel used toda. areer Links L S S Z O N E. O M 32. raw each letter that has at least one line of setr and sketch its line(s) of setr. Which letters have one line of setr? Which letters have two lines of setr? Word Reflections eterine if the entire word has an lines of setr. If so, write the word and draw the line(s) of setr Kaleidoscope esigns ind the easure of the angle between the irrors (a) that produces the kaleidoscope design. Use the equation a 8 0. n hapter 5 ongruent Triangles

8 Page 8 of 9 Student elp LOOK K or help with the distance forula, see p. 94. EXMPLE Show that TcTJKL. Show that the corresponding sides are congruent. or sides on a horizontal grid line, subtract the -coordinates. 5 4 LJ 5 4 or sides on a vertical grid line, subtract the -coordinates. Show Triangles are ongruent KL 4 ( 2) 2 2 or an other sides, use the distance forula. (5 ) 2 ( 4 2 ) JK (5 ) 2 ( ( 4 ( 2 )) ( 2 ) the SSS ongruence Postulate, T c TJKL. (, 2) (5, 2) J (, 2) (5, 4) L(5, 2) K (5, 4) Showing Triangles are ongruent In Eercises 40 and 4, refer to the eaple above. Show that TcTE (2, 3) (3, 4) (2, ) (6, ) (2, ) (6, ) E(, ) (, ) (3, ) E(2, 3) ( 4, 3) (, 3) Standardized Test Practice 42. Multiple hoice Which triangle shows the iage when TXYZ is reflected in the -ais? E L J TE TPQR TJKL None of these Z K R X Y ΠP 43. Multiple hoice ow an lines of setr does the figure at the right have? 0 2 J Reflections and Setr 289

9 Page 9 of 9 Mied Review Showing Lines are Parallel ind the value of so that p q. (Lesson 3.5) p 82 p q ( 0) q p 92 (3 ) q inding ngle Measures ind the easure of a. (Lesson 4.2) lgebra Skills oparing Nubers opare the two nubers. Write the answer using >, <, or. (Skills Review, p. 662) and and and and and and. Quiz 3. Sketch the overlapping triangles separatel. Mark all congruent angles and sides. Which postulate or theore can ou use to show that the triangles are congruent? (Lesson 5.5) Use the diagra to find the indicated easure(s). (Lesson 5.6) 2. ind. 3. ind ML and JK. 4. ind J L M 9 25 K eterine the nuber of lines of setr in the figure. (Lesson 5.7) hapter 5 ongruent Triangles