Grade 10 / vectors in the plane 65 exercises corrected

a
ChingQuizz : 6 exercises available for Quizz assessment : 1. Revisions transformations E. 867 The hatched figure is ob-tained by applying a plane transformation to the white figure. In each case : Specify the type of transformation (axial symmetry, cen-tral symmetry, translation, rotation) . Show and specify le (s) élément (s) caractéristique (s) of this transformation (axis, center, angle, direction of ro-tation) E. 9338 In the figure below, each of the quadrilaterals quad1 , quad2 and quad3 is the image of the quadrilateral TRAP by a transformation. Copy the three sentences below onto your copy and complete, without justification, each of them with the number of one of the transformations proposed in the following table : a The quadrilateral quad1 is the image of the quadrilateral TRAP by transformation number . . . b The quadrilateral quad2 is the image of the quadrilateral TRAP by transformation number . . . c The quadrilateral quad3 is the image of quadrilateral TRAP by transformation number . . . Transformation number 1: translation which trans-forms the point D into the point E . Transformation number 2: rotation of center A and angle 90 o counterclockwise. Transformation number 3: central symmetry of center D . Transformation number 4: translation which trans-forms the point E into the point D . Transformation number 5: rotation of center A and angle 120 o counterclockwise. Transformation number 6: axial symmetry of axis ( DE ) . E. 866 ABCO , CDEO , EFGO and GHAO are squares shown below. BDFH is a square with center O . 1 a What is the image of the triangle ABC by the or-thogonal symmetry of axis ( GC ) ? b What is the image of triangle ABC by the rotation of center O , of angle 90 o that brings E into C ? 2 Using transformations whose characteristic elements (center of symmetry, axis of symmetry, . . . ) will be spec-ified, copy and complete the following sentences without justifying the answer. a triangle GFE is the image of ABC triangle by . . . b triangle OCD is the image of triangle ABC by . . . 2. Introduction to translation E. 2761 Consider the figure below : https://chingmath.fr quizzChap/709 chapExoCorrec/867 sacados/867 abcdef chapExoCorrec/9338 sacados/9338 TPRAquad3DEquad2quad1 chapExoCorrec/866 sacados/866 ABCDEFGHO chapExoCorrec/2761 sacados/2761
1 The hatched ovoid figure was obtained by a translation of the white ovoid figure. Show a vector characterizing this translation. 2 The hatched polygon was obtained by a translation of the white polygon. Draw three representatives of this translation. 3 Make a conjecture about these two translations. E.2762 The drawing below shows two fig-ures, one white and the other hatched : 1 It is assumed that the white figure is the symmetrical of the hatched figure by a translation. a Draw precisely two vectors, of your choice, of this translation. b Connect the two origins of these two vectors and con-nect their two ends. 2 a Justify that this quadrilateral is not a parallelogram. b Can we conclude that these two figures are symmetri-cal to each other by translation? E. 2763 In the grid below, consider the translation T of vector −→ u : 1 Draw the image A of the point A by the translation of vector ~u . 2 Plot the image of rectangle BCDE by translation T . 3 Draw the translate of polygon FGHIJ by vector −→ u . 3. First notions E. 5987 Complete each box in the table below with the words ˇ iden-tical ı, ˇ différent ı or ˇ opposé ı : Par rapport à −→ u comparaison de la direction du sens de la longueur −→ v −→ w −→ r −→ s −→ t https://chingmath.fr chapExoCorrec/2762 sacados/2762 chapExoCorrec/2763 sacados/2763 uABCDEFGHIJ chapExoCorrec/5987 sacados/5987 uvwrst
E. 493 In the grid below : 1 Draw a representative of the vector −→ u having as origin the point A . 2 Draw a representative of the vector −→ u having as its ex-tremity the point B . 3 Draw a vector −→ v of the same length as −→ u , but different from −→ u . 4 Draw a vector −→ w same direction, same sense as −→ u , but different from −→ u . 5 Draw a vector −→ s of the same direction and length as −→ u , but different from −→ u . E.11323 Consider the diagram below : Using the points shown in the diagram: 1 a Name a vector that has the same direction and sense as vector −−→ MA but a different length. b Name a vector that is opposite to vector −−→ MA . 2 a Name a vector with the same direction but a differ-ent length than vector −−→ QN . b Name a vector opposite to vector −−→ QN . E.11839 Complete each box in the table below with the words ˇ iden-tical ı, ˇ différent ı or ˇ opposé ı : Par rapport à −→ u comparaison de la direction du sens de la longueur −→ v −→ w −→ r −→ s −→ t 4. Equal vectors E. 928 ABCD is a square with center O . The points E , F , G , H are the middles of the sides of the square. 1 What is the image of point B by the counterclockwise rotation of center O , of angle 90 o ? 2 What is the image of point E by the vector translation −−→ OC ? 3 Complete the dotted lines to verify the equalities: a −→ AO = −−−→ O : : : = −−−→ : : : G b −−→ FC = −−−→ : : : H c −−→ CG = −−−→ O : : : = −−→ : : : A https://chingmath.fr chapExoCorrec/493 sacados/493 uuAB chapExoCorrec/11323 sacados/11323 ABCDEFGHIJKLMNOPQRST chapExoCorrec/11839 sacados/11839 uvwrst chapExoCorrec/928 sacados/928 AFBEOGDHC
E. 11322 The figure below is composed of 15 squares. Copy and complete the dotted lines correctly to check the equalities: a −→ SO = −−−→ P : : : = −−−→ : : : D b −→ V I = −−→ R : : : = −−→ : : : A E. 11334 Consider the tiling of the plane below composed of equilateral triangles. 1 Copy and complete to verify the equalities: −→ LI = −−−→ F : : : = −−−→ : : : O 2 With respect to vector −−→ GD : a Give a vector that has the same direction but not the same length. b Give a vector that has the same length but not the same direction. c Give a vector that is opposite to vector −−→ GD . E. 11853 On considère le pavage du plan con-stitué d’hexagone régulier représentée ci-dessous : 1 Sans justification, compléter les pointillées pour vérifier les égalités: a −−→ GD = −−→ L : : : b −→ GJ = −−−→ D : : : c −−→ CP = −−→ A : : : 2 Sans justification, compléter les pointillées pour vérifier les égalités: a −−→ OP = −−→ : : : L b −−→ EP = −−−→ : : : G c −−→ HF = −−→ : : : T E. 11866 On considère le pavage du plan con-stitué de parallélogrammes représentée ci-dessous : 1 Sans justification, compléter les pointillées pour vérifier les égalités: a −→ GR = −−−→ C : : : b −→ IK = −−−→ E : : : 2 Sans justification, compléter les pointillées pour vérifier les égalités: a −→ RG = −−→ : : : A b −−→ MJ = −−−→ : : : D 5. Vector plot on white paper E.9339 In the plane, consider the three points A , B , M shown below : https://chingmath.fr chapExoCorrec/11322 sacados/11322 ABCDEFGHIJKLMNOPQRSTUVWX chapExoCorrec/11334 sacados/11334 ABCDEFGHIJKLMNOPQRST chapExoCorrec/11853 sacados/11853 CDEFABHGIJKLMONPQRSUTV chapExoCorrec/11866 sacados/11866 ABCDEFGHIJKLMNOPQRST chapExoCorrec/9339 sacados/9339 ABM
Consider the two distances : r = AB ; r = AM 1 a Draw the circle C with center M and radius r . b Draw the circle C with center B and radius r . 2 a Among the two points of intersection of circles C and C , note N as the point such that quadrilateral ABNM is a parallelogram. b Justify that vectors −−→ AB and −−→ MN are equal. 3 Which of the four properties that characterize a paral-lelogram can be used to justify the answer to question 2 a Property 1: if the diagonals of a quadrilateral intersect at their midpoints, then the quadrilateral is a parallelogram. Property 2: if a quadrilateral has opposite sides parallel to each other, then that quadrilateral is a parallelogram. Property 3: If a quadrilateral has opposite sides of equal length, then that quadrilateral is a parallelogram. Property 4: If two opposite sides of a quadrilateral are parallel and of equal length, then that quadrilateral is a parallelogram. E.2773 Method : to draw the representative of a vector with a given origin. Consider the configuration below : Place point N so that vectors −−→ AB and −−→ MN are equal. E.9340 Example: Consider the configuration below : Drawings are to be made with a straightedge and a compass : 1 a Place the point D image of the point C by the vector translation −−→ AB . b What is the nature of the quadrilateral ABDC ? 2 a Place the point E such that ACBE is a parallelo-gram. b Characterize the translation transforming point B into E . E.2764 Consider the translation T of the plane which transforms the point A into B : Tracings must be made using a straightedge and compass : 1 Place the point D , image of the point C by the transla-tion that transforms A into B . 2 Place the point F , image of the point E by the transla-tion of the vector −−→ AB . 3 Place the point G such that G has the point H as its image by the translation of vector −−→ AB . 6. Equal Vectors and Parallelograms E.8101 For each of the propositions below, state whether it is true or false. (no justification required) a The vectors −−→ AB and −−→ CD are equal. The quadrilateral ABCD is a parallelogram. b The segments [ AB ] and [ CD ] have the same midpoint I . The quadrilateral CBDA and a parallelogram c The quadrilateral MNPQ is a parallelogram. The vec-tors −−→ MN and −−→ QP are equal. d The quadrilateral WXY Z is a parallelogram. The diag-onals [ WX ] and [ Y Z ] have the same midpoint. https://chingmath.fr chapExoCorrec/2773 sacados/2773 r662-1 ABM chapExoCorrec/9340 sacados/9340 r662-1 ABC chapExoCorrec/2764 sacados/2764 ABCEH chapExoCorrec/8101 sacados/8101
E.9582 1 a Draw the point C image of the point A by the vector translation −−→ MN . b Draw point D image of point B by vector translation −−→ MN . 2 What is the nature of the quadrilateral ACDB ? Justify your answer. E.7513 Consider two parallelograms ABCD and ABFE , a representation of which is given below : 1 a Justify the vector equality: −−→ AB = −−→ DC b Justify vector equality: −−→ DC = −−→ EF 2 Deduce the nature of the quadrilateral DCFE . E.8102 Complete the blanks to make each sentence correct: a If −→ AI = −−→ :::::: then the point I is the midpoint of the seg-ment [ AB ] . b If ABCD is a parallelogram then −−→ AB = −−→ :::::: c If K is the middle of segment [ XY ] then −−→ :::K = −−→ :::::: d If −−→ MN = −−→ PQ then : : : : : : : : : is a parallelogram. E.918 1 Draw a triangle ABC rectangular in B . 2 Place the point T such that : −−→ AB = −→ CT . What is the nature of the quadrilateral ABTC ? 3 Place the point M such that : −−→ BC = −−→ MT . Justify that the quadrilateral BCTM is a rectangle. E. 11870 Dans le plan, on considère les trois points A , B , C : 1 Placer le point D tel que ABCD soit un parallélo-gramme. 2 Placer le point E tel que ABEC soit un parallélogramme. 7. Vector and plane geometry E.9341 In the plane, consider the triangle ABC shown below : 1 a Place point M , which is the image of point B under the symmetry with center C . b Place point N , the image of point M , by translation of vector −−→ AB . 2 Determine the nature of quadrilateral ABNM . Justify your answer. E.7917 Consider the regular hexagon ABCDEF shown opposite. 1 Justify that the triangle COB is equilateral. 2 Justify that the points F , O and C are aligned. 3 Demonstrate that the straight lines ( BC ) and ( EF ) are parallel. 4 Justify that the vectors −−→ BC and −−→ FE are equal. https://chingmath.fr chapExoCorrec/9582 sacados/9582 MNAB chapExoCorrec/7513 sacados/7513 ABCDEF chapExoCorrec/8102 sacados/8102 chapExoCorrec/918 sacados/918 chapExoCorrec/11870 sacados/11870 ABC chapExoCorrec/9341 sacados/9341 ABC chapExoCorrec/7917 sacados/7917 OABCDEF
E.7512 Consider a triangle ABC any and the points M , N , P respective middles of the sides [ AC ] , [ AB ] , [ BC ] : 1 Justify that the straight lines ( BC ) and ( MN ) are par-allel. 2 a What can be said about the vectors −−→ CP and −−→ MN ? Justify your answer. b Justify that the quadrilateral MNPC is a parallelo-gram. 8. Introduction to the sum of vectors: composition of translations E.8123 Proposition-Definition: If, to any point of the plane, we apply a translation of vec-tor −→ u , then a translation of vector −→ v , we obtain a new translation whose vector is −→ u + −→ v . This new translation is called the composed of the vector translation −→ u by the vector translation −→ v Consider the six vectors shown below : 1 a Place the point N image of the point M by the vec-tor translation −→ u . b Place the point P image of the point N by the vector translation −→ v . c Draw a vector −→ w representing the composition of the vector translation −→ u by the vector translation −→ v . 2 a Place the point S image of the point R by the trans-lation of the vector −→ u 2 . b Draw a representative of the vector −→ u 2 + −→ v 2 . 3 Draw a representative of the vector −→ u 3 + −→ v 3 having origin U . E.925 Eight pairs of vectors are shown be-low. For each of these pairs, draw a representative of the sum of its two vectors : https://chingmath.fr chapExoCorrec/7512 sacados/7512 ABCMNP chapExoCorrec/8123 sacados/8123 −u−vM−u2−v2R−u3−v3U chapExoCorrec/925 sacados/925 −u1−v1−u2−v2−u3−v3−u4−v4−u5−v5−u6−v6−u7−v7−u8−v8
E.931 For each of the sentences below, complete the dotted lines correctly: 1 The compound of the translation of vector −→ EI by that of vector −−→ FG is a translation of vector −−−→ E : : : 2 The compound of the vector translation −→ JG by that of vector −→ JB is a vector translation −−→ J : : : 3 The compound of the translation of vector −−→ GF by that of vector −−→ GH then that of −→ EI is a vector translation . . . . . . 4 The compound of the translation of vector −−→ CH by that of vector −→ CJ then that of vector −−→ BH is a translation of vector . . . . . . E.2783 In the plane, consider the two vec-tors −→ u and −→ v as well as the two points A and D shown below : Note: All constructions must be done using a non-graduated ruler and a compass. 1 Draw a representative of the vector −→ u + −→ v with origin at point A . 2 Consider point E obtained by the image of point D through the composite of the translation of vector −→ u and the translation of vector −→ v . Place point E . E.933 A , B and C are three points in the plane. Reproduce a figure similar to the one below and complete it as you go along: Hint: constructions will be made using a straightedge and compass. 1 Construct the point M image of A by the vector transla-tion −−→ BC . 2 Give a vector equal to the vector −−→ MA . 3 Construct the point K such that : −→ CA + −−→ CB = −−→ CK 4 Demonstrate that : −−→ MA = −−→ AK . What can be said for the point A ? 9. Sum of vectors E.11315 In the grid below, consider the three vectors −→ u , −→ v , −→ w and the three points A , C , E shown below : https://chingmath.fr chapExoCorrec/931 sacados/931 ABCDJHIGEF chapExoCorrec/2783 sacados/2783 uvAD chapExoCorrec/933 sacados/933 ABC chapExoCorrec/11315 sacados/11315 −u−v−wACE
1 Place point B , the image of point A , by translating vec-tor −→ u + −→ v . 2 Place point D , the image of point C , by translating vec-tor −→ v + −→ w . 3 Place point F , the image of point E , by translating vec-tor −→ u + −→ w . E.9580 In the grid below, consider the three vectors −→ u , −→ v , −→ w and the two points A , C shown below : 1 Place point B , the image of point A , by translating vec-tor −→ u + −→ v . 2 Place point D , the image of point C , by translating vec-tor −→ v + −→ w . E.9343 In the grid below, consider the three vectors −→ u , −→ v , −→ w and the three points A , C , E shown below : 1 Place point B , the image of point A , by translating vec-tor −→ u + −→ v . 2 Place point D , the image of point C , by translating vec-tor −→ v + −→ w . 3 Place point F , the image of point E , by translating vec-tor −→ u + −→ v + −→ w . E.11335 In the grid below, consider the three vectors −→ u , −→ v , −→ w and the three points A , C shown below : 1 Place point B , the image of point A , by translating vec-tor −→ u + −→ v . 2 Place point D , the image of point C , by translating vec-tor −→ u + −→ v + −→ w . 3 Place point E such that the image of point E by vector translation −→ v + −→ w is point F . 10. Sum of vectors: representing E.11313 The figure below is composed of 15 squares. Copy and complete the equations below correctly: https://chingmath.fr chapExoCorrec/9580 sacados/9580 −u−v−wAC chapExoCorrec/9343 sacados/9343 −u−v−wACE chapExoCorrec/11335 sacados/11335 −u−v−wACF chapExoCorrec/11313 sacados/11313 ABCDEFGHIJKLMNOPQRSTUVWX
a −−→ NJ + −→ JQ = −−−→ N : : : b −−→ NW + −−→ WI = −−−→ P : : : c −−→ HO + −−→ PE = −−−→ H : : : d −−→ NH + −→ JU = −−−→ N : : : e −−→ V N + −→ NI = −−−→ : : : D f −→ QI + −→ EJ = −−→ : : : T E.11314 Consider the drawing below : Copy and complete the dotted lines appropriately : a −−→ CN + −→ LF = −−−→ B : : : b −−→ QF + −→ RT = −−−→ M : : : c −→ OL + −−→ NH = −−→ : : : A d −−→ PH + −→ AC = −−−→ : : : E E.11325 Replace ".. . "with the de-sired point : 1 −→ FI + −−→ FG = −−−→ F : : : 2 −→ JG + −→ JE = −−→ J : : : 3 −→ IH + −−→ HE = −−−→ : : : F 4 −→ FI + −→ BI = −−−→ : : : E E.11324 Consider the tiling of the plane consisting of regular hexagons shown below : Copy and complete the equations : a −−→ BE + −−→ OR = −−−→ B : : : b −−→ CK + −−→ SK = −−−→ : : : C c −−→ PE + −−→ CU = −−→ I : : : d −→ SE + −−→ OG = −−−→ : : : B 11. Chasles’ relationship E.11677 The figure below is composed of 15 squares. Copy and complete the equations below correctly: a −−→ NJ + −→ JQ = −−−→ N : : : b −−→ NW + −−→ WI = −−−→ P : : : c −−→ HO + −−→ PE = −−−→ H : : : d −−→ NH + −→ JU = −−−→ N : : : e −−→ V N + −→ NI = −−−→ : : : D f −→ QI + −→ EJ = −−→ : : : T E.9344 Consider the grid below and the 10 points shown. 1 a Using the points on the figure, name all vectors equal to the vector −−→ FE . b Use the question to give a representative of the vector −→ AE + −−→ FG . 2 Use the Chasles relationship to answer the following ques-tions : a −−→ FE + −−→ FH + −→ JB b −→ IH + −−→ FD + −→ JE c −−→ DF + −→ IG + −−→ HJ d −−→ DG + −→ EA + −−→ DC https://chingmath.fr chapExoCorrec/11314 sacados/11314 ABCDEFGHIJKLMNOPQRST chapExoCorrec/11325 sacados/11325 ABCDJHIGEF chapExoCorrec/11324 sacados/11324 CDEFABHGIJKLMONPQRSUTV chapExoCorrec/11677 sacados/11677 ABCDEFGHIJKLMNOPQRSTUVWX chapExoCorrec/9344 sacados/9344 ABCDJHIGEF
E.9346 The figure below is made up of 15 squares. Using the Chasles relationship, copy and correctly complete the equalities below : a −−→ NJ + −−→ BO = −−−→ N : : : b −−→ GC + −→ CJ + −→ JO = −−−→ G : : : c −−→ PE + −→ DL = −−−→ : : : Q d −−→ PH + −−→ HK + −−→ KV = −−−→ : : : V E.506 Replace ".. . "with the de-sired point : 1 −−→ EF + −→ EA = −−−→ C : : : 2 −→ HI + −−→ ED + −−→ FG = −−→ J : : : 3 −−→ HG + −→ HI + −−→ FB = −−−→ : : : B E.2784 Consider the drawing below : Copy and fill in the blanks : a −→ BI + −−→ NC = −−−→ K : : : b −−→ QF + −→ JL = −−−→ O : : : c −−→ NH + −→ OL = −−−→ : : : F d −−→ PH + −→ GI + −→ JI = −−→ L : : : E.9345 Consider the tiling of the plane con-sisting of regular hexagons shown below : 1 Without justification, give a representative of each of the proposed sums : a −−→ AD + −→ LR + −→ DI b −−→ HF + −−→ BG + −−→ BG 2 Without justification, complete the dotted lines correctly to verify the equality: a −−→ DB + −−→ GK + −→ : : : = −−→ DP b −→ : : : + −−→ BE + −−→ KO = −−→ MO E.496 Let ABCD be a parallelogram. Let I (resp. J ) be the midpoint of the segment [ AB ] (resp. of the segment [ DC ] ) . In each case, determine a representative of the resulting vec-tor using the points of this configuration : a −→ AC + −→ JA b −→ AI + −−→ AD c −→ AC + −−→ BD + −→ JB E.11678 Consider the grid below and the 10 points shown. 1 a Using the points on the figure, name all vectors equal to the vector −−→ FE . b Use the question to give a representative of the vector −→ AE + −−→ FG . 2 Use the Chasles relationship to answer the following ques-tions : a −−→ FE + −−→ FH + −→ JB b −→ IH + −−→ FD + −→ JE c −−→ DF + −→ IG + −−→ HJ d −−→ DG + −→ EA + −−→ DC 12. Commutativity of the sum of vectors https://chingmath.fr chapExoCorrec/9346 sacados/9346 ABCDEFGHIJKLMNOPQRSTUVWX chapExoCorrec/506 sacados/506 ABCDJHIGEF chapExoCorrec/2784 sacados/2784 ABCDEFGHIJKLMNOPQRST chapExoCorrec/9345 sacados/9345 CDEFABHGIJKLMONPQRSUTV chapExoCorrec/496 sacados/496 ABCDIJ chapExoCorrec/11678 sacados/11678 ABCDJHIGEF
E.8118 In the plane, consider the points A and B and the vectors −→ u and −→ v shown below : 1 a Construct the point A image of the point A by the vector translation −→ u . b Construct the point A  image of the point A by the vector translation −→ v . c Construct a vector representative −→ u + −→ v 2 a Construct the point B image of the point B by the vector translation −→ v . b Construct the point B  image of the point B by the vector translation −→ u . c Construct a vector representative −→ v + −→ u . 3 Compare the two vectors −→ u + −→ v and −→ v + −→ u . E.929 1 For each of the quadrants below : a Place the point B translated from the point A by the vector translation −→ u . b Draw point C translated from point B by vector trans-lation −→ v . In each dial, the point C obtained is called the translate of the point A by the vector −→ u + −→ v . 2 In the first quadrant : a Place the point B translated from the point A by the vector −→ v . b Place the point C translated from the point B by the vector −→ u . c What can youyou say about the translation composed of translations of vectors −→ u then that of −→ v and of the compound translation of the translations of vectors −→ v and ~u ? 13. Sum of vectors, Chasles relation and commutativity E.936 Complete the dotted lines in the following questions by choos-ing a vector from the graph above : a −→ b + −→ a = −→ : : : b −→ d + −→ j = −→ : : : c −→ c + −→ d = −→ : : : d −→ : : : + −→ i = −→ f e −→ ‘ + −→ i = −→ : : : + −→ e E.932 Copy the statement onto your copy and complete the dotted lines : a −−→ BC + −−→ GF + −−→ HE = −−−→ E : : : b −−→ GF + −→ JC + −−→ GH = −−→ A : : : c −−→ CF + −→ IC + −−→ FB = −−→ I : : : d −−→ HG + −→ HI + −−→ FB = −−−→ G : : : https://chingmath.fr chapExoCorrec/8118 sacados/8118 −u−vAB chapExoCorrec/929 sacados/929 11234AuvAuvAuvAuv chapExoCorrec/936 sacados/936 −k−a−g−j−f−h−d−c−e−b−i−‘ chapExoCorrec/932 sacados/932 ABCDJHIGEF
E.930 middle East - Eastern Group -June 2005 For this exercise, complete the figure given below. Three points A , B , and C were placed. 1 Construct the point E such that ABEC is a parallelo-gram. 2 a Construct the point F such that : −−→ BF = −−→ BA + −−→ BC . b What is the nature of the quadrilateral ABCF ? No justification is required. 3 Show that : −−→ FC = −−→ CE . What can be inferred for the point C ? E.6545 The figure below is made up of 15 squares. Copy the vector equalities below and correctly complete the blanks with the missing point : a −−→ NJ + −−→ BO = −−−→ N : : : b −→ OL + −−→ AK + −→ QI = −−→ : : : R c −→ TI + −−→ : : : J = −→ JF d −−→ PH + −−→ OD + −−−→ C : : : = −−→ V K E.924 The figure opposite is made up of identical regular hexagons : Complete the dotted lines, detailing your calculations if possible : a −−→ BE + −→ DL + −−→ KG b −−→ AD + −−→ FM + −→ LG 14. Chasles relation and decomposition E.8317 The figure below is composed of squares : 1 a Justify that : −→ OI = −−→ NP + −−→ DC b Give two vectors −→ ¸ and −→ ˛ such that : −−→ PD = −→ ¸ + −→ ˛ c Establish equality: −→ OI + −−→ PD = −−→ NC 2 For each question, give a representative vector of the form : a −−→ DO + −−→ HS b −→ OI + −→ AR E.494 The figure opposite is made up of three regular hexagons. 1 a Justify equality: −−→ IM = −→ IJ + −→ JE + −−→ ED + −−→ DM b Give three vectors −→ ¸ , −→ ˛ , −→ ‚ such that : −→ CJ = −→ ¸ + −→ ˛ + −→ ‚ c Deduce a vector representing the sum : −→ CJ + −−→ IM 2 Using a method similar to the previous question, deter-mine a representative of the sum : −→ LB + −−→ GM https://chingmath.fr chapExoCorrec/930 sacados/930 ABC chapExoCorrec/6545 sacados/6545 ABCDEFGHIJKLMNOPQRSTUVWX chapExoCorrec/924 sacados/924 CDEFABMLKJIHG chapExoCorrec/8317 sacados/8317 ABCDEFGHIJKLMNOPQRST chapExoCorrec/494 sacados/494 CDEFABMLKJIHG
15. Opposite vectors E.6996 Consider the parallelogram ABCD shown below and the point O intersection of its diagonals. 1 Name a vector opposite the −−→ BC vector. 2 Name a vector opposite the vector −−→ OB originating at point O . 3 Cite a vector opposite the vector −−→ AD having as its end point the point B . E.6997 In the plane, consider a point O and a vector −−→ AB shown below : 1 Using the compass and unsharpened straightedge, place the points A and B symmetrical to the points A and B with respect to the point O . 2 What can we say about the vectors −−→ AB and −−−→ A B ? E.9579 The figure below is composed of squares : Determine a representative of each of the following sums : a −→ DI + −−→ QO b −−→ DQ − −−→ DB c −−→ DQ + −−→ KB + −→ IC 16. Share E.935 The figure below shows a rect-angle ABCD with center O and point E , symmetrical to O about C . 1 Consider the rotation of center O that transforms B into C . What is the image of D by this rotation? (No justi-fication is required.) 2 Of the following statements, copy the ones that are true (no justification is required) . −→ OA = −−→ OC −−→ OC = −−→ OE OA = CE −−→ BE = −−→ BO + −−→ OE −−→ AB + −→ AC = −−→ BC −−→ AB + −−→ AD = −→ AC D is the image of C by the vector translation −−→ AB 17. Unclassified exercises https://chingmath.fr chapExoCorrec/6996 sacados/6996 ABCDO chapExoCorrec/6997 sacados/6997 ABO chapExoCorrec/9579 sacados/9579 ABCDIJKLOMNPQ chapExoCorrec/935 sacados/935 ABCDOE
E.934 1 Draw a square EFGH with side 4 cm . 2 Place the point J such that : −→ FJ = −−→ EF 3 Place the point K such that : −−→ FK = −−→ EH + −−→ EF E.9810 Consider, in the plane, the two vectors −→ u and −→ v below : 1 Draw in the grid a representative −→ w of the sum −→ u + −→ v . 2 Draw in the grid a representative −→ y of the difference −→ u −−→ v . E.5341 Let ABCD be a parallelogram. Con-sider the points I , J , K and L defined by: −→ AI = x · −−→ AD ; −→ DJ = x · −−→ DC ; −−→ CK = x · −−→ CB ; −→ BL = x · −−→ BA où x is a real number belonging to the interval 0 ; 1 . Demonstrate that the quadrilateral IJKL is a parallelogram. https://chingmath.fr chapExoCorrec/934 sacados/934 chapExoCorrec/9810 sacados/9810 −u−v chapExoCorrec/5341 sacados/5341 ABCDIJKL