Grade 9 / Rotation, homothety and similar triangles 32 exercises (100% corrected)

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image PDF : /home/_data/myLocaltexmf/tex/sensRotation-1.pdf
image PDF : /home/_data/myLocaltexmf/tex/sensRotation-2.pdf
image PDF : /home/_data/myLocaltexmf/tex/sensRotation-2.pdf
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121110987654321121110987654321121110987654321ParisLeCairePraia 12111098765432112111098765432113:0015:20 OABCDEF ABPQABAB ChingQuizz : 9 exercises available for Quizz assessment : 1. Introduction to rotation E.6847 1 The capitals Paris (France) , Cairo (Egypt) , Praia (Cape Verde) are located on different time zones. In a company, three clocks are used to know the current time in each of these cities: a What is the angle of rotation to take the small hand of the Paris clock to the position of the small hand in Cairo? b What is the angle of rotation to take the small hand of the Praia clock to the position of the small hand of Paris? 2 Aissata looks at her watch at two different times : a What is the angle of the rotation made by the large needle during these two moments? b What is the angle of the rotation made by the small needle during these two moments? E.6849 In the configuration below, consider the two triangles ABC and DEF . 1 a Draw three arcs of a circle AD , CF , BE of center O . b Measure the three angles ∠ AOD , ∠ BOE and ∠ COF . c Copy and complete the following sentence : The triangle ABC has as its image the triangle DEF by the rotation of center . . . . . . , angle . . . . . . and di-rection . . . . . . 2 a Using the compass, compare the measurements of the segments : [ AB ] and [ DE ] ; [ AC ] and [ DF ] b Copy and complete the following sentence : If two segments are the image of each other by a rota-tion then they are . . . . . . . . . 3 a Using the protractor, compare the measures of the angles : ∠ ABC et ∠ DEF ; ∠ BCA et ∠ DEF b Copy and complete the following sentence : If two angles are the image of each other by a rotation then they are . . . . . . 2. Characteristic of a rotation E.10706 Consider the 8 points shown below where : the rotation R 1 has transformed the segment [ AB ] into the segment [ A B ] . rotation R 2 transformed segment [ A B ] into segment [ A  B  ] . https://chingmath.fr chapExoCorrec/6847 sacados/6847 121110987654321121110987654321121110987654321ParisLeCairePraia 12111098765432112111098765432113:0015:20 chapExoCorrec/6849 sacados/6849 OABCDEF chapExoCorrec/10706 sacados/10706 ABPQABAB
ABCDPQ ABCDEFGHIJKLMPQRS O32AB36o11098765413121120191817161514droite(d Determine the characteristics of the rotations R 1 and R 2 . E.10768 Consider the configuration below : 1 Consider the rotation R 1 of center P which transforms the point A into the point C . a Give the characteristics of the rotation R 1 . b Is segment [ CD ] the image of segment [ AB ] by rotation R 1 ? Justify your answer. 2 Consider the rotation R 2 of center Q which transforms the point B into the point C . a Give the characteristics of the rotation R 2 . b Is segment [ DC ] the image of segment [ AB ] by rotation R 2 ? Justify your answer. 3. Recognize the image by a rotation E.9294 We are interested in the blades of a decorative garden windmill. They are shown in the figure below Given : BCDE , FGHI , JKLM , and PQRS are rectangles that can be stacked on top of each other. C , B , A , J , K on the one hand, and G , F , A , P , Q on the other hand are aligned. AB = AF = AJ = AP 1 What transformation allows us to go from rectangle FGHI to rectangle PQRS ? 2 What is the image of rectangle FGHI when rotated coun-terclockwise around center A by angle 90 o ? E.10767 Consider the following figure, composed of twenty patterns numbered from 1 to 20 , built on a regular decagon (ten-sided polygon) in which : AOB = 36 o 1 Give the image of pattern 20 by symmetry of axis the line ( d ) . 2 Give the image of pattern 7 by central symmetry of cen-ter O . 3 Give the characteristics of the rotation R 1 by which pat-tern 3 is the image of pattern 1 ? 4 Give the characteristics of the rotation R 2 by which pat-tern 11 is the image of pattern 19 ? https://chingmath.fr chapExoCorrec/10768 sacados/10768 ABCDPQ chapExoCorrec/9294 sacados/9294 ABCDEFGHIJKLMPQRS chapExoCorrec/10767 sacados/10767 O32AB36o11098765413121120191817161514droite(d
ABCDOEFGHIJK12345678 011xyAB 011xyAB E.9295 Hint: in this exercise, no justification is required A square ABCD . has been constructed We constructed the point O on the line ( DB ) , outside the segment [ DB ] and such that : OB = AB . The point H is the symmetrical of D with respect to O . The figure below was obtained by repeatedly using the same rotation of center O and angle 45 o . The resulting figure is symmetrical about the ( DB ) axis and about the O point. 1 Give two different squares, images of each other by axial symmetry of axis ( DB ) . 2 Is the square 3 the image of the square 8 by central sym-metry of center O ? 3 Consider the rotation of center O that transforms the square 1 into the square 2 . What is the image of square 8 by this rotation? 4 Consider the rotation with center O that transforms the square 2 into the square 5 . Specify the image of segment [ EF ] by this rotation. 4. Rotation and marker E.10770 In the plane provided with a refer-ence frame, consider the segment [ AB ] shown below : 1 We note [ A B ] the image of the segment [ AB ] by the rotation of center O , of angle 90 o clockwise. Represent the segment [ A B ] in the above reference frame. 2 Give the coordinates of the points A and B . E.10708 In the plane provided with a refer-ence frame, consider the segment [ AB ] shown below : 1 We note [ A B ] the image of the segment [ AB ] by the rotation of center O , of angle 90 o clockwise. Represent the segment [ A B ] in the above reference frame. 2 Give the coordinates of the points A and B . https://chingmath.fr chapExoCorrec/9295 sacados/9295 ABCDOEFGHIJK12345678 chapExoCorrec/10770 sacados/10770 011xyAB chapExoCorrec/10708 sacados/10708 011xyAB
011xyAB 011xyAB 011xyAB 011xyAB AT1T2BQ1Q2OF1F2 E.10710 In the plane with a reference point, consider the segment [ AB ] shown below : 1 We note [ A B ] the image of the segment [ AB ] by rota-tion around the center O , angle 90 o counterclockwise. Represent the segment [ A B ] in the coordinate system above. 2 Give the coordinates of points A and B . E.10709 In the plane provided with a refer-ence frame, consider the segment [ AB ] shown below : 1 Note [ A B ] the image of the segment [ AB ] by the rota-tion of center O , of angle 90 o clockwise. Represent the segment [ A B ] in the above reference frame. 2 Give the coordinates of the points A and B . E.10711 In the plane provided with a refer- ence frame, consider the segment [ AB ] shown below : 1 We note [ A B ] the image of the segment [ AB ] by the ro-tation of center O , of angle 90 o in the counterclockwise direction. Represent the segment [ A B ] in the above reference frame. 2 Give the coordinates of the points A and B . E.10769 In the plane provided with a refer-ence frame, consider the segment [ AB ] shown below : 1 We note [ A B ] the image of the segment [ AB ] by the ro-tation of center O , of angle 90 o in the counterclockwise direction. Represent the segment [ A B ] in the above reference frame. 2 Give the coordinates of the points A and B . 5. Rotation with the grid E.6848 Consider the six figures : 1 Copy and complete the following sentences : a The figure F 2 is the image of F 1 by the rotation r 1 with center . . . . . . and angle . . . . . . b Figure T 2 is the image of T 1 by the rotation r 2 of center . . . . . . and angle . . . . . . c Figure Q 2 is the image of Q 1 by the rotation r 3 of center . . . . . . and angle . . . . . . 2 Specify the direction of each rotation. This can be done by: or indicate by a diagram / the direction of rota-tion or specify that the rotation is clockwise or counter-clockwise . https://chingmath.fr chapExoCorrec/10710 sacados/10710 011xyAB chapExoCorrec/10709 sacados/10709 011xyAB chapExoCorrec/10711 sacados/10711 011xyAB chapExoCorrec/10769 sacados/10769 011xyAB chapExoCorrec/6848 sacados/6848 AT1T2BQ1Q2OF1F2
OABCPDEFG OABCDEFG OABC PDEFG E.6850 In the grid below, are represented the triangle ABC and the parallelogram DEFG . 1 Draw the triangle A B C image of the triangle ABC by the rotation of center O , angle 90 o and direction . 2 Draw the parallelogram D E F G image of the parallel-ogram DEFG by the rotation of center P , angle 90 o and direction clockwise. E.6863 Consider below the triangle ABC and the quadrilateral DEFG shown below : The proposed grid has the following characteristics : All circles are concentric with center O and their radii have measure 0.5 cm , 1 cm , 1.5 cm . . . lines are concurrent at the point O and two consecutive lines form an angle of 10 o . 1 Draw the triangle A B C image of the triangle ABC by the rotation of center O , of angle 70 o in the direction . 2 Draw the quadrilateral D E F G image of the quadri-lateral DEFG by the rotation of center O , angle 30 o clockwise. 6. Rotation on white paper E.6851 Consider the triangle ABC shown below : Draw the triangle A B C image of triangle ABC by the rota-tion of center O , angle 60 o and direction . E.9298 Consider the quadrilateral DEFG shown below : Draw the quadrilateral D E F G image of the quadrilateral DEFG by the clockwise rotation of center P , angle 45 o . https://chingmath.fr chapExoCorrec/6850 sacados/6850 OABCPDEFG chapExoCorrec/6863 sacados/6863 OABCDEFG chapExoCorrec/6851 sacados/6851 OABC chapExoCorrec/9298 sacados/9298 PDEFG
OABC ABAB ABABCDCD E.10771 Consider the triangle ABC shown below : Draw the triangle A B C image of triangle ABC by the rota-tion of center O , angle 60 o and direction . 7. Rotation and search for the center E.9308 As an airplane approaches the air-port and aligns itself with the runway, it performs a rotation. To study the center of this rotation, a portion of the wings was modeled by a segment : the segment [ AB ] before the ap-proach phase and the segment [ A B ] during the approach phase : Hint: be A a point in the plane and A its image by a rotation. The center O of this rotation belongs to the per-pendicular bisector of segment [ AA ] . 1 a Draw the perpendicular bisector ( d ) of segment [ AA ] . b Draw the median ( d ) of the segment [ BB ] . c Name O the intersection of the two mediators ( d ) and ( d ) . 2 a Check the length equalities: OA = OA ; OB = OB b What remains to be verified to establish that the plane has rotated about the point O ? E.9299 In the plane, consider the quadrilat-eral ABCD shown below : Consider the quadrilateral A B C D obtained from the quadrilateral ABCD by a rotation. Only the [ A B ] image of the [ AB ] segment was made correctly. The rest of the quadrilateral was drawn by hand. Determine the center O of the rotation as well as the angle of the rotation. E.10774 The graph below shows two boats that are rotated in relation to each other. Determine the center of this rotation. https://chingmath.fr chapExoCorrec/10771 sacados/10771 OABC chapExoCorrec/9308 sacados/9308 ABAB chapExoCorrec/9299 sacados/9299 ABABCDCD chapExoCorrec/10774 sacados/10774
OABAB C1C2C3C4C5 IxyT1T3T2T4T5 E.10707 Below, consider the segment [ AB ] and the point O : 1 Consider the rotation R 1 with center O such that : The image of the point A is the point A The image of the point B is the point B Determine the characteristics of the rotation R 1 2 Determine the characteristics of rotation R 2 achieving: The image of the point A is the point B The image of the point B is the point A 8. All transformations E.6856 We consider the following 5 chairs : The chairs C 2 , C 3 , C 4 , and C 5 are obtained from the chair C 1 from a transformation. Specify the nature and characteristic elements of each of these transformations. 9. All transformations and grids E.6830 The triangles T 2 , T 3 , T 4 , and T 5 are obtained from the triangle T 1 using a transformation of the plane : Without justification, copy and complete the sentences below : 1 The triangle T 2 is the image of the triangle T 1 by sym-metry having . . . . . . 2 The image of the triangle T 1 by axial symmetry with axis . . . . . . is the triangle . . . . . . 3 triangle T 1 has as its image the triangle T 5 by . . . . . . 4 triangle T 4 is obtained, from the T 1 triangle, by . . . . . . Characteristic elements of each of the transformations will be given https://chingmath.fr chapExoCorrec/10707 sacados/10707 OABAB chapExoCorrec/6856 sacados/6856 C1C2C3C4C5 chapExoCorrec/6830 sacados/6830 IxyT1T3T2T4T5
uIxyABC E.6831 Consider the triangle ABC , the line ( xy ) , the point I and the vector −→ u represented in the grid below : 1 Draw the triangle DEF image of the triangle ABC by axial symmetry of axis ( xy ) . 2 Draw triangle GHI image of triangle ABC by central symmetry of center I . 3 Trace triangle JKL image of triangle ABC by vector translocation −→ u 4 Draw triangle MNO image of triangle ABC by clockwise rotation of center I , angle 90 o . 10. Friezes and tessellations with symmetry, translation and rotation E.6853 Consider the frieze below : 1 Circle the pattern used to construct this frieze by succes-sive translations. 2 What transformation is used to go from the elementary pattern on the left to the pattern on the right below : 3 Starting with the elemental pattern of this frieze , summarize in order the different transformations used to construct this curve. E.6854 Consider the frieze below : 1 Circle the pattern used to construct this frieze by succes-sive translations. 2 What transformations are used to go from the elementary pattern on the left to the pattern on the right below : 3 Starting with the elemental pattern of this frieze , summarize in order the different transformations used to construct this curve. E.6852 Consider the frieze below : 1 Circle the pattern used to construct this frieze by succes-sive translations. 2 What transformation is used to move from the pattern on the right to the pattern on the left below : 3 Starting with the elemental pattern of this frieze , summarize in order the different transformations used to construct this curve. https://chingmath.fr chapExoCorrec/6831 sacados/6831 uIxyABC chapExoCorrec/6853 sacados/6853 chapExoCorrec/6854 sacados/6854 chapExoCorrec/6852 sacados/6852
121110987654321121110987654321TempsATempsB E.6841 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.6844 We consider the paving below : The colors of the paving pieces will be ignored. 1 Starting with the black pattern, specify the transforma-tions needed to reconstruct this tiling. 2 Below is shown an elementary pattern (left) and a basic pattern (right) of the tiling: Quelle (s) transformation (s) est (/are) necessary to obtain the basic pattern from an elementary pattern? 11. Unclassified exercises E.10773 Consider the same clock photographed at two different times : Determine the angle of rotation undergone by the hour hand between time A and time B . https://chingmath.fr chapExoCorrec/6841 sacados/6841 chapExoCorrec/6844 sacados/6844 chapExoCorrec/10773 sacados/10773 121110987654321121110987654321TempsATempsB