Grade 8 / Translations 32 exercises (100% corrected)

a
ABC(dO M1M2M3M4 123456789101112131415161718192021222324 ABC1725634 ChingQuizz : 2 exercises available for Quizz assessment : 1. Reminders and introduction to translation E.9022 Consider triangle ABC , point O , and line ( d ) shown below : 1 Draw triangle DEF , which is the image of triangle ABC under the axial symmetry with axis ( d ) . 2 Draw triangle GHI , the image of triangle ABC , by cen-tral symmetry with center O . E.6846 We consider the paving below : 1 Name a transformation to turn the pattern M 1 into M 2 . 2 Name a transformation to turn the pattern M 1 into M 3 . 3 Describe a transformation whose pattern image M 1 and pattern M 4 . The colors of the paving pieces will not be considered. 2. Use of the translation E.7933 The figure below represents a paving whose basic pattern is a regular hexagon. Some of these hexagons have been numbered : What is the image of the hexagon 14 by the translation that transforms the hexagon 2 into the hexagon 12 ? E.9011 Consider the seven dogs shown be-low all numbered and the three points A , B , C : Complete the blanks in the following sentences : a The dog 3 is the image of the dog 6 by the translation transforming the point A into . . . . . . b The dog 5 is the image of the dog . . . by the translation transforming the point C into B . c The dog 4 is the image of the dog . . . by the translation transforming the point A to B . https://chingmath.fr chapExoCorrec/9022 sacados/9022 ABC(dO chapExoCorrec/6846 sacados/6846 M1M2M3M4 chapExoCorrec/7933 sacados/7933 123456789101112131415161718192021222324 chapExoCorrec/9011 sacados/9011 ABC1725634
-224-1250261272283294305316327338349351036113712381339144015411642174318441945204621472248234924502551525354555657585960616263 BACDO1234 OABCD15 ABCDEFGHIJKLMNOPQRST 172839410511612181319142015211622172329243025312632273328344035413642374338443945514652475348544955505662576358645965606661 P1P2P3ABCDEFG d The dog . . . is the image of the dog 7 by the translation transforming the point C to A . E.9009 Consider the frieze shown below : 1 List all the patterns present in the figure that are ob-tained by translating the pattern 14 shown in gray. 2 List all the patterns present in the figure that are ob-tained by translating the pattern 16 . E.7934 In this exercise, the square ABCD is not represented at full size. No justification is expected for questions 1 and 2 . 1 Consider the square ABCD of center O shown opposite, divided into four su-perimposable polygons, numbered 1 , 2 , 3 and 4 . What is the image of the polygon 1 by the central symmetry of center O ? 2 The figure below is part of the paving, one basic pattern of which is the ABCD square from question 1 . What transformation from polygon 1 yields polygon 5 ? E.8995 Consider the drawing below : 1 By the translation that tranforms the point L into N : a what is the image of segment [ LG ] ? b what is the image of segment [ CK ] ? 2 By the translation that tranforms the point S into M : a what is the image of segment [ GI ] ? b what is the image of segment [ NH ] ? E.9145 Consider the tiling of the plane shown below with regular hexagons, some of which have been numbered : 1 a By the translation transforming the hexagon 11 into the hexagon 32 , what is the image of the hexagon 31 ? b By the translation transforming the hexagon 49 into the hexagon 41 , what is the image of the hexagon 21 ? 2 By the translation transforming the hexagon 51 into the hexagon 42 , what is the hexagon that has as its image the hexagon 26 ? 3. Translation with grid E.8998 Consider in the plane provided with a grid of three parallelograms and 7 points shown below : https://chingmath.fr chapExoCorrec/9009 sacados/9009 -224-1250261272283294305316327338349351036113712381339144015411642174318441945204621472248234924502551525354555657585960616263 chapExoCorrec/7934 sacados/7934 Extrait Antilles Guyanne Septembre 2020 BACDO1234 OABCD15 chapExoCorrec/8995 sacados/8995 ABCDEFGHIJKLMNOPQRST chapExoCorrec/9145 sacados/9145 172839410511612181319142015211622172329243025312632273328344035413642374338443945514652475348544955505662576358645965606661 chapExoCorrec/8998 sacados/8998 P1P2P3ABCDEFG
Q1CD T1ABCD T1ABCD F1AB ABAB 1 Consider the parallelogram P 2 to be the image by trans-lation of the parallelogram P 1 . What is the image of the point A by this translation? 2 Consider that the parallelogram P 1 has as its image the parallelogram P 3 by a translation. What is the image of the point A by this translation? E.9010 In the grid below, the quadrilateral Q 1 and the two points C and D : Draw the image of the quadrilateral Q 1 by the translation transforming the point C into D . E.6834 In the grid below, consider the tri-angle T 1 and the four points A , B , C and D : 1 a Draw the triangle T 2 obtained by the translation of the triangle T 1 that transforms the point A into B . b Draw the triangle T 3 obtained by the translation of the triangle T 1 that transforms the point C into D . 2 Which transformation gets the triangle T 2 into the tri-angle T 3 ? Put two points image of each other by this transformation. E.9023 In the grid below, consider the tri-angle T 1 and the four points A , B , C and D : 1 a Draw the triangle T 2 obtained by the translation of the triangle T 1 that transforms the point A into B . b Draw the triangle T 3 obtained by the translation of the triangle T 2 that transforms the point C into D . 2 Which transformation gets the triangle T 1 into the tri-angle T 3 ? Put two points image of each other by this transformation. E.6835 In the grid below, consider the figure F 1 and the two points A and B : Draw the image of figure F 1 by the translation that trans-forms point A into B . 4. Translation and parallelogram E.6832 In a ski resort, a cable car moves on its cable : Let’s try to describe the movement made by the cabin at two different times. 1 Answer the following questions : The two representations of the cabin are: a symmetrical by axial symmetry? If so, specify the axis of symmetry? b symmetrical by a central symmetry? If yes, specify the center of symmetry. 2 Let us highlight two points A and B and their images A and B ’: a Compare segment [ AB ] and its image [ A B ] . b Give the nature of the quadrilateral ABB A . https://chingmath.fr chapExoCorrec/9010 sacados/9010 Q1CD chapExoCorrec/6834 sacados/6834 T1ABCD chapExoCorrec/9023 sacados/9023 T1ABCD chapExoCorrec/6835 sacados/6835 F1AB chapExoCorrec/6832 sacados/6832 ABAB
ABMP ABCD ABC MNABC E.9012 Consider the 4 points A , B , M and N shown below : 1 a Place the point I middle of the segment [ MB ] . b Place the point N symmetrical to the point A with respect to the point I . 2 a Place the point J middle of the segment [ PB ] . b Place the point Q symmetrical to the point A with respect to the point J . 3 a Draw the triangles AMP and BNQ . b What can be said about the transformation turning the triangle AMP into BNQ . 4 Explain the method used to construct the symmetric of point M by the translation transforming point A into B . 5. Translation on white paper E.6838 Proposition: be A , B , C three points of the plane. We denote D the image of point C by the tanslation that trans-forms A into B . Construction method : We’ll use the property of the diagonals of a parallelogram: they intercept at their middles. To plot the point D image of the point C by the translation that transforms A into B : we place the point I midway along segment [ AB ] . we place the point D image of the point A by central symmetry of center I . In the plane, consider the five points shown below : Constructions will be made using the graduated ruler and compass 1 a Place the point I middle of the segment [ AN ] . b Draw the image of point A by the transformation that turns M into N . 2 a Place the point J middle of the segment [ BN ] . b Draw the image of point B by the translation that transforms M into N . 3 Finish the construction to draw the triangle A B C im-age of the triangle ABC by the translation transforming the point M into N https://chingmath.fr chapExoCorrec/9012 sacados/9012 ABMP chapExoCorrec/6838 sacados/6838 ABCD ABC MNABC
ABC(dORS ABCDMN A−u−v−w E.9021 Consider the triangle ABC , the three points O , R , S and the line ( d ) shown below : 1 Draw the triangle DEF image of the triangle ABC by axial symmetry of axis ( d ) . 2 Draw the triangle GHI image of the triangle ABC by central symmetry of center O . 3 Draw the triangle JKL image of the triangle ABC by the transformation that turns the point R into S . E.6839 In the plane, consider the quadrilat-eral ABCD and the vector −→ u shown below Constructions will be made using the graduated ruler and compass Construct the symmetric of the quadrilateral ABCD by vec-tor translation −→ u . E.6840 Below are shown the point A and three vectors −→ u , −→ v and −→ w : 1 a Place the point A symmetrical to the point A by the vector translation −→ u . b Place the point A  symmetrical of the point A by the vector translation −→ v . c Place the point A  symmetrical to the point A  by the vector translation −→ w . 2 What do we notice? 6. Frieze, tiling and translation E.6842 We consider the paving below : https://chingmath.fr chapExoCorrec/9021 sacados/9021 ABC(dORS chapExoCorrec/6839 sacados/6839 ABCDMN chapExoCorrec/6840 sacados/6840 A−u−v−w chapExoCorrec/6842 sacados/6842
Motif 1Motif 2ABCABCD ABCD Starting from the black pattern, specify the transformations necessary to reconstruct this tiling. The colors of the pieces of the paving will be ignored. E.6843 We consider the paving below : Starting from the black pattern, specify the transformations necessary to reconstruct this tiling. The colors of the pieces of the paving will be ignored. E.6845 We consider the paving below : Starting from the black pattern, specify the transformations necessary to reconstruct this tiling. The colors of the pieces of the paving will be ignored. E.6855 Frame, in the frieze below, the pat-tern that is repeated successively by translation for its con-struction. E.9024 Below is a representation of a tessel-lation of the plane : This tiling can be achieved by choosing a pattern (consisting of several ˇ carreaux ı) and applying the same translations to it several times. Show a possible pattern as well as the translations used to reproduce that pattern to obtain the tiling. 7. Translation and plane geometry E.7967 Gaspard is working with dy-namic geometry software to build a frieze. He constructed a triangle ABC isosceles in C (pattern 1) and then got the rhombus ACBD (pattern 2) . Here are the screenshots of his work. 1 Specify a transformation to complete the pattern 1 to obtain the pattern 2 . 2 Once the pattern 2 was constructed, Gaspard repeatedly applied translation. This gives him the frieze below. Specify which translation it is : 8. Properties of translation https://chingmath.fr chapExoCorrec/6843 sacados/6843 chapExoCorrec/6845 sacados/6845 chapExoCorrec/6855 sacados/6855 chapExoCorrec/9024 sacados/9024 chapExoCorrec/7967 sacados/7967 Motif 1Motif 2ABCABCD ABCD
MNF1F2ABCDEF ABCDEFGHIJKLMNO 45oFigure 1 Figure 2 Figure 3 E.6836 In the plane, consider the three dis-tinct points A , B and C . Note the point D image of the point C by the translation transforming the point A into B . Without justification, answer the following questions with true or false : 1 Segments [ AC ] and [ BD ] have the same measurement. 2 segments [ AD ] and [ BC ] have the same measure. 3 The quadrilateral ABCD is a parallelogram. 4 The segments [ AD ] and [ BC ] have the same middle. E.6837 In the plane, consider the two fig-ures F 1 and F 2 whose F 2 is the image of F 1 by the translation transforming M into N . 1 a Verify that the segments [ FM ] and [ NC ] have the same midpoint. b What is the nature of the quadrilateral MNFC ? 2 Name, at least, three other parallelograms present using these figures. 3 Copy and complete the following sentences : a If two lines are translationally symmetrical then these two lines are . . . . . . b If two segments are translationally symmetric then these two segments are . . . . . . c If two angles are translationally symmetric then these two angles are . . . . . . d If A has image B and if C has image D by the same translation then the quadrilateral ABDC is a . . . . . . In particular, the segments [ AD ] and [ BC ] have . . . . . . e If two circles are symmetric by translation then the centers are . . . . . . and the radii have . . . . . . E.8996 Consider the frieze shown below : This frieze is constructed from the pattern of the pattern in figure 1 ; from this pattern, figure 2 is constructed, and then figure 3 is constructed to finally obtain the frieze. 1 a Which transformation takes the figure 1 (the pat-tern) to 2 . b Which transformation moves from figure 2 to figure 3 . c By the translation from figure 3 to the frieze, give the image of point D . 2 What is the nature of the quadrilateral AFHG ? Justify your answer. 9. Translation and theorem E.6833 Consider the triangle ABC such that : AB = 6.8 cm ; BC = 5.1 cm ; AC = 8.5 cm 1 a Draw the triangle ABC . b Construct the point D translated from the point A by the translation that transforms the point B into C . c Justify that the quadrilateral ABCD is a parallelo-gram. 2 a Establish that the triangle ABC is right-angled at B . b What is the nature of the quadrilateral ABCD ? Jus- tify your answer. E.9013 Consider the triangle ABC right-angled A such that : AB = 5 cm ; AC = 6 cm 1 a Draw the triangle ABC . b Place the point M image of the point B by the trans-lation that transformed A to B . c Place the point N image of the point C by the trans-lation that transformed A into C . 2 Establish that the lines ( BC ) and ( MN ) are parallel. 10. In addition: translation and other transformations https://chingmath.fr chapExoCorrec/6836 sacados/6836 chapExoCorrec/6837 sacados/6837 MNF1F2ABCDEF chapExoCorrec/8996 sacados/8996 ABCDEFGHIJKLMNO 45oFigure 1 Figure 2 Figure 3 chapExoCorrec/6833 sacados/6833 chapExoCorrec/9013 sacados/9013
ABCOO AB(d(d E.1012 1 a Draw A the symmetric of A by central symmetry of center O . Then plot A  the symmetrical of A by the symmetry of center O . b What can we say about the straight lines ( OO ) and ( AA  ) ? What can we say about the lengths OO and AA  ? Which theorem can be used to affirm the previous observation? 2 Draw the successive symmetries of point B then of point C by symmetry of center O then of center O . 3 What is the transformation from triangle ABC to trian-gle A  B  C  ? E.8997 In the plane, consider the two straight lines ( d ) and ( d ) parallel and the points A and B shown below : 1 a Draw the points A and B symmetrical to the points A and B , respectively, by the axis symmetry ( d ) . b Draw the points A  and B  symmetric to the points A and B respectively by the axis symmetry ( d ) . 2 a The segments [ AB ] and [ A  B  ] are parallel and of equal length. How can we establish by geometric rea-soning? b What transformation can be used to transform the seg-ment [ AB ] into [ A  B  ] ? https://chingmath.fr chapExoCorrec/1012 sacados/1012 ABCOO chapExoCorrec/8997 sacados/8997 AB(d(d