Grade 9 / Theorem of Thales 60 exercises (100% corrected)

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GHJIKVWXYZ ABCMNABCMN2cmABCMNABCMN2cm6cmABCMNABCMN2cmABCMNABCMN2cm6cm3cm JesaisJ’utiliseJ’endéduisLes pointsA,M,Bsont alignés.Les pointsA,N,Csont alignés.(MN==(BCD’aprèslethéorèmedeThalès,onal’égalité des quotients:AMABANACMNBCChaînonsdéductifs TZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cmTZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cm5cm 1. Reminders: Thales’ theorem E.5021 Solve the following equations : a 7 x = 21 4 b 15 8 = x 9 c x 32 = 5 8 E.11449 Thales’ theorem: If points A , B , M are aligned, points A , C , N are aligned, and if lines ( BC ) and ( MN ) are parallel, then : AM AB = AN AC = MN BC Note: the equality of the length ratios reflects the propor-tionality of the lengths between the ˇ nested ı triangles when two of their sides are parallel. We have represented two configurations of Thales where ( GH ) == ( IJ ) and ( XY ) == ( V W ) . For each of these triangles, write the equality of quotients obtained by using Thales’ theorem. E.11450 Commented example: Consider the triangle ABC shown below : Points M and N belong to segments [ AB ] and [ AC ] , re-spectively, and are such that lines ( MN ) and ( BC ) are parallel. We have the numerical application : 2 6 = 3 AC = MN BC We will use the equality: 2 6 = 3 AC According to the cross product, we have : 2 × AC = 6 × 3 2 × AC = 18 AC = 18 2 AC = 9 cm Consider the triangle TXZ shown below and the points L and I belonging respectively to the segments [ TZ ] and [ TX ] such that ( IL ) == ( XZ ) . 1 Complete the deductive chain below : https://chingmath.fr chapExoCorrec/5021 sacados/5021 chapExoCorrec/11449 sacados/11449 GHJIKVWXYZ chapExoCorrec/11450 sacados/11450 ABCMNABCMN2cmABCMNABCMN2cm6cmABCMNABCMN2cmABCMNABCMN2cm6cm3cm JesaisJ’utiliseJ’endéduisLes pointsA,M,Bsont alignés.Les pointsA,N,Csont alignés.(MN==(BCD’aprèslethéorèmedeThalès,onal’égalité des quotients:AMABANACMNBCChaînonsdéductifs TZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cmTZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cm5cm
JesaisJ’utiliseJ’endéduisLes points ..., ..., ...sont alignés.Les points ..., ..., ...sont alignés.:::==:::D’après le théorème de Thalès, on a l’égalité desquotients:Chaînonsdéductifs EFGHI1;532 MGPUA4;5m3;5m5;6m7;2m JesaisJ’utiliseJ’endéduisLes pointsM,A,Pet les pointsM,U,Gsont alignés dans le même ordre.MAMPMUMGD’aprèslaréciproqueduthéorèmedeThalès:(AU==(PGChaînonsdéductifs ATNCR3;2cm4;8cm7;5cm5cm JesaisJ’utiliseJ’endéduisLes points ..., ..., ...et les points ..., ..., ...sont alignés dans le même ordre.::::::::::::D’après la réciproque du théorème de Thalès::::==:::Chaînonsdéductifs 2 Use the equality of quotients in question 2 to determine the length of segment [ TX ] . E.9309 In the plane, we consider the config-uration below : The straight lines ( FG ) and ( HI ) are respectively parallel to each other. 1 Give the length of segment [ EH ] . 2 Using Thales’ theorem, determine the length of segment [ EI ] . 3 Deduce that the length of segment [ GI ] . E.10757 Solve the following equations : a x x + 1 = 1 3 b x + 2 3 x − 4 = 2 3 c 2 x x − 3 = 4 E.10792 1 Consider the three numbers : a = 2 5 ; b = 2 3 ; c = 3 Determine the solution to the equation : a b = x c 2 Consider the three numbers : a = 4 3 ; b = 5 2 ; c = 14 3 Determine the solution to the equation : x a = b c 2. Reminders: reciprocal of Thales’ theorem E.11451 Commented example: Consider the triangle MGP shown below and the points A and U belonging respectively to the segments [ MP ] and [ MG ] : MA =3 ; 5 m ; MP =5 ; 6 m ; MU =4 ; 5 m ; AT =7 ; 2 m We have the numerical application : MA MP = 3 ; 5 5 ; 6 = 0 ; 625 ; MU MG = 4 ; 5 7 ; 2 = 0 ; 625 Consider triangle ANT and the two points R and C belong-ing respectively to segments [ AN ] and [ AT ] . We have the measurements : AN =7 ; 5 cm ; AR =4 ; 8 cm ; AC =3 ; 2 cm ; AT =5 cm 1 Prove that the quotients are equal: AR AN = AC AT 2 Complete the deductive chain below : https://chingmath.fr JesaisJ’utiliseJ’endéduisLes points ..., ..., ...sont alignés.Les points ..., ..., ...sont alignés.:::==:::D’après le théorème de Thalès, on a l’égalité desquotients:Chaînonsdéductifs chapExoCorrec/9309 sacados/9309 EFGHI1;532 chapExoCorrec/10757 sacados/10757 chapExoCorrec/10792 sacados/10792 chapExoCorrec/11451 sacados/11451 MGPUA4;5m3;5m5;6m7;2m JesaisJ’utiliseJ’endéduisLes pointsM,A,Pet les pointsM,U,Gsont alignés dans le même ordre.MAMPMUMGD’aprèslaréciproqueduthéorèmedeThalès:(AU==(PGChaînonsdéductifs ATNCR3;2cm4;8cm7;5cm5cm JesaisJ’utiliseJ’endéduisLes points ..., ..., ...et les points ..., ..., ...sont alignés dans le même ordre.::::::::::::D’après la réciproque du théorème de Thalès::::==:::Chaînonsdéductifs
ABCMN0;91;21;5 4cm12cm5cm7;5cmABCDO KGHJIVWXYZ ABCMN1;552 ABCOMND ABCDG E.3462 We consider the configuration below representing a Thales configuration. Moreover, we know that : AN =3.5 . Show that the following lines ( BC ) and ( MN ) are parallel. E.9310 In the configuration below, A ∈ [ OC ] and B ∈ [ OD ] . Show that the lines ( AB ) and ( CD ) are parallel: 3. Butterfly configuration E.2167 We have represented two configura-tions of Thales where ( GH ) == ( IJ ) and ( XY ) == ( V W ) . In each case, list the equalities of length ratios given by Thales’ theorem : E.3451 In the plane, we consider the config-uration : The straight lines ( BC ) and ( MN ) are respectively parallel to each other. Using Thales’ theorem, determine the length of segment [ AN ] . E.660 Consider the lines ( BD ) and ( CN ) intersecting O with A a point of ( CN ) and M a point of ( BD ) such that : The lines ( BN ) , ( AM ) and ( CD ) are parallel to each other. 1 Give all length ratios equal to : a OM OD b OB OM 2 We are given the following measurements : NO = 5 cm ; OA = 4 cm ; OM = 3 cm AC = 2 cm ; CD = 3 cm a Determine the measure of segment [ BO ] . b Determine the length AM . Then, derive the length BN . 4. Problem and theorem of Thales E.5695 A folding stool was geometrically modeled by segments [ CB ] and [ AD ] for the metal frame and segment [ CD ] for the fabric seat. We have : CG = DG =30 cm , AG = BG =45 cm and AB = 51 cm . For comfort, the seat [ CD ] is parallel to the floor represented by the line ( AB ) . Determine the length CD of the seat. Hint: leave all traces of research apparent. Even if the work is not completed, it will be considered in the scoring. https://chingmath.fr chapExoCorrec/3462 sacados/3462 ABCMN0;91;21;5 chapExoCorrec/9310 sacados/9310 4cm12cm5cm7;5cmABCDO chapExoCorrec/2167 sacados/2167 KGHJIVWXYZ chapExoCorrec/3451 sacados/3451 ABCMN1;552 chapExoCorrec/660 sacados/660 ABCOMND chapExoCorrec/5695 sacados/5695 ABCDG
RivièreADRBV20m12m15mLa∏guren’estpasàl’échelle FBCRGDiamètre75cmhauteurduregard:1;80mhauteurdurebord:1mépaisseur du mur:20cm ACBNMACBNM5;2cmACBNMACBNM5;2cm3cmACBNMACBNM5;2cmACBNMACBNM5;2cm3cm7cm AMNBC3;2cm2;1cm7;4cm E.6282 Indication : any trace of response, even if incomplete, will be taken into account in the evaluation Joachim has to cross a river with a group of friends. He wants to install a rope so that the insecure people can hold on. He wants to know the width of the river at this location (named D ) to determine if the rope he has is long enough. To do this, he has located a tree (named A ) on the other bank. He travels 20 meters on the straight bank o where he is and finds a new landmark : a rock (named R ) . Then he continues for 12 meters and then moves away from the river, at right angles, until the rock is aligned with the tree from his vantage point (named B ) . To do this, he travels 15 m . He is now satisfied : his 30 -meter-long rope is long enough for him to install it between the points D and A . Using the figure, confirm your decision. E.5049 A young shepherd stands at the edge of a cylindrical well with a diameter of 75 cm : he aligns his gaze with the lower edge of the well and the bottom of the well to estimate its depth. The bottom of the well and the rim are horizontal. The well is vertical. 1 Using the diagram below (not to scale) , give the lengths CB , FG , RB in meters. 2 Calculate the depth BG of the well. 3 The shepherd notices that the water level in the well is 2.60 m . The young shepherd needs 1 m 3 of water to water all his sheep. Will he find enough in this well? Hint: we will use : ı ≈ 3.1416 5. Thales’ theorem and equation E.11446 Solve the following equations : a x x + 8 = 2 3 b x − 5 x = 7 8 E.11447 Solve the equations : a x + 8 x + 12 = 4 5 b x − 5 x + 2 = 4 5 E.1141 1 Solve the following equation : ( E ) : x x + 5.2 = 3 7 2 In the figure at right. the straight lines ( MN ) and ( BC ) are parallel. Determine the measure of the segment [ AM ] . E.1136 In the figure below, the lines ( BC ) and ( MN ) are parallel. Determine the measure of the segment [ AC ] . https://chingmath.fr chapExoCorrec/6282 sacados/6282 RivièreADRBV20m12m15mLa∏guren’estpasàl’échelle chapExoCorrec/5049 sacados/5049 Brevet Pondichery Avril 2012 Le volume d'un cylindre ne fait pas partie du socle commun FBCRGDiamètre75cmhauteurduregard:1;80mhauteurdurebord:1mépaisseur du mur:20cm chapExoCorrec/11446 sacados/11446 chapExoCorrec/11447 sacados/11447 chapExoCorrec/1141 sacados/1141 ACBNMACBNM5;2cmACBNMACBNM5;2cm3cmACBNMACBNM5;2cmACBNMACBNM5;2cm3cm7cm chapExoCorrec/1136 sacados/1136 AMNBC3;2cm2;1cm7;4cm
ABCDMN5;5cm3;5cm1;5cm ABCEFx223 6cm9cm4cm1cmABCDJIK ABCD45m24mIM E.11454 Let ABCD be a rectangle and M a point belonging to the segment [ CD ] . Let N be the point of intersection of the lines ( AM ) and ( BC ) . We have the following measurements : AB =5 ; 5 cm ; BC =3 ; 5 cm ; CN =1 ; 5 cm Let x be the length of the segment [ DM ] . 1 Establish the equality: x 5 ; 5 − x = 3 ; 5 1 ; 5 2 Deduce the length DM . E.658 The figure opposite was made freehand ; we have the follow-ing properties : the point E belongs to the line ( AB ) ; point F belongs to line ( AC ) ; the straight lines ( EF ) and ( BC ) are parallel. Determine the value of ˇ x ı. E.655 Consider a parallelogram ABCD such that : AD = 6 cm ; CD = 4 cm ; AC = 9 cm Let J be the point on segment [ AB ] that satisfies : AJ = 1 cm . The line parallel to line ( AD ) passing through point J inter-sects lines ( AC ) and ( CD ) at points I and K , respectively. The figure below illustrates this situation : 1 a Prove that lines ( IJ ) and ( BC ) are parallel. b Find the length of segment [ AI ] . 2 a Find the length of segment [ KC ] . b Use this to find the length of segment [ IJ ] . Let x denote the length of segment [ IJ ] . E.10842 1 Solve the equation : x 51 − x = 1 2 2 Consider a rectangular park with dimensions 45 m × 24 m where two paths intersect (represented by paths [ BD ] and [ IC ] ) : a Establish that the path represented by segment [ BD ] measures 51 m . b The two paths intersect at point M . Determine the distance DM . 6. Theorem of Thales, equations and Pythagorean theorem E.1139 1 a Draw a rectangle ABCD such that : AB = 1.5 cm ; AD = 6 cm b Place the point I belonging to [ BC ] such that : BI = 1 3 BC . c Name M the point of intersection of the lines ( AI ) and ( CD ) . 2 Determine the length of segment [ MC ] . 3 Determine the length of segment [ AM ] . https://chingmath.fr chapExoCorrec/11454 sacados/11454 ABCDMN5;5cm3;5cm1;5cm chapExoCorrec/658 sacados/658 ABCEFx223 chapExoCorrec/655 sacados/655 6cm9cm4cm1cmABCDJIK chapExoCorrec/10842 sacados/10842 ABCD45m24mIM chapExoCorrec/1139 sacados/1139
ABCDE600m270m450m 4m4;8m5m6mABCDO 34;5710;5MNBCA PQZFG5;533;31;8 E.11500 A farmer wants to cultivate a field represented by the triangle ABC shown opposite. The figure, which is not to scale, provides the following informa-tion : triangle ABC is a right tri-angle at B ; triangle CDE is a right triangle at D ; points C , E , and A are collinear; points C , D , and B are collinear; AB =600 m ; BC =450 m ; CD =270 m 1 Show that the segment [ AC ] measures 750 meters. 2 a Show that the lines ( ED ) and ( AB ) are parallel. b Show that segment [ DE ] measures 360 meters. 3 Show that the area of triangle CDE is 48 600 m 2 . E.5780 On a full moon night, Romeo wishes to visit Juliet. He has a ladder of 10 m in length. The windowsill is at a height 4.8 m but there is a wall be-tween it and the house : this wall is 50 cm thick, 4 m high. The driveway separating the wall from the house has a width of 1 m Will Romeo manage to put the end of the ladder on Juliet’s windowsill? 7. Reciprocal of Thales’ theorem in butterfly configuration E.666 For the configuration below, show that the straight lines ( AB ) and ( CD ) are parallel: E.653 Show that the lines ( BC ) and ( MN ) are parallel: E.9311 Consider the configura-tion below representing Thales configurations. Show that the straight lines ( FG ) and ( PQ ) are parallel. 8. Thales’ theorem and reciprocal in two butterfly configurations https://chingmath.fr chapExoCorrec/11500 sacados/11500 ABCDE600m270m450m chapExoCorrec/5780 sacados/5780 chapExoCorrec/666 sacados/666 4m4;8m5m6mABCDO chapExoCorrec/653 sacados/653 34;5710;5MNBCA chapExoCorrec/9311 sacados/9311 PQZFG5;533;31;8
AFGEDCB ABCMNPR ABCDOFE2;5cm3cm5cm2;4cm4cm ABCGF E.671 The unit of length is the cen-timeter In the figure opposite, which is not full-scale, the straight lines ( BC ) and ( GF ) are parallel. We know that : AB = 3 ; CE = 2.4 ; AC = 4 ; BD = 1.8 BC = 4.5 ; AF = 3.6 1 Calculate length GF . 2 Are the straight lines ( BC ) and ( ED ) parallel? Justify. E.659 For each of the two questions in this exercise, specify the course property used The figure to the right is not shown at full size. The straight lines ( BC ) and ( MN ) are parallel. The following lengths are given : AB =2.4 cm ; AC =5.2 cm ; AN =7.8 cm ; MN =4.5 cm 1 Calculate the lengths AM and BC . 2 Knowing that AP =2.6 cm and AR =1.2 cm , show that the lines ( PR ) and ( BC ) are parallel. 9. Thales’ theorem and reciprocal in a nested and butterfly configuration E.670 In the figure opposite, the straight lines ( AC ) and ( DB ) are parallel. 1 Calculate the length AO . 2 Are the straight lines ( EF ) and ( OD ) parallel? Justify your assertion. E.5184 Hint: the figure below is not full-scale, it is not to be re-produced. The points A , C and F are aligned, as are the points B , C and G . The straight lines ( AB ) and ( GF ) are parallel. The following measurements are given : AB = 3 cm ; FC = 8.4 cm ; FG = 11.2 cm 1 Calculate the length CA . 2 Let D be the point on segment [ CF ] and E be the point on segment [ GF ] such that : FD = 6.3 cm ; FE = 8.4 cm Show that the straight lines ( GC ) and ( ED ) are parallel. https://chingmath.fr chapExoCorrec/671 sacados/671 Groupe Nord - Septembre 2002 - 4 points AFGEDCB chapExoCorrec/659 sacados/659 Groupe Sud - Juin 2003 - 5 points ABCMNPR chapExoCorrec/670 sacados/670 ABCDOFE2;5cm3cm5cm2;4cm4cm chapExoCorrec/5184 sacados/5184 ABCGF
ABCDEFG ABCDEF ABCDEO OABMN E.661 The unit is centimetres. In the figure below, the straight lines ( AB ) and ( CD ) are parallel. The straight lines ( AD ) and ( BC ) intersect at E . We give : DE =6 , AE =10 , AB =20 and BE =16 . Figures are not full-scale. They are not to be reproduced. 1 Calculate distance CD . 2 The points F and G belong to the segments [ BC ] and [ AB ] respectively. They verify: BF = 12.8 ; BG = 16 . Show that the straight lines ( FG ) and ( AE ) are parallel. E.3461 The following figure is not a full-scale figure. The unit of length is centimeters ; we give : AB = 8 ; BC = 9 ; AC = 6 ; AE = 4 1 The lines ( DE ) and ( BC ) are parallel. Compute AD . Its exact value will be given, and then its value rounded to the tenth of a centimeter. 2 Let F be the point such that C , B , and F are aligned in that order, with BF =6 . Show that the lines ( EF ) and ( AB ) are parallel. E.664 1 a Draw a triangle ABC such that : AC = 7.5 cm ; BC = 10 cm ; AB = 6 cm . b Place E on [ AC ] such that AE = 4.5 cm and F on [ BC ] such that BF = 6 cm . 2 Are the straight lines ( AB ) and ( EF ) parallel? Justify. 3 We draw the line parallel to ( AB ) passing through C . This line intersects ( BE ) at L . Determine CL . 10. Thales’ theorem and reciprocal in the same configuration E.673 The figure below shows a dia-gram of an ironing board. The segment [ AB ] and [ EC ] represent the legs. The straight lines ( AB ) and ( EC ) intersect at O . We give : AD = 125 cm ; AC = 100 cm ; OA = 60 cm OB = 72 cm ; OE = 60 cm ; OC = 50 cm 1 Show that the line ( AC ) is parallel to the line ( EB ) . 2 Calculate the distance EB in cm . E.9312 The structure of a stool is shown op-posite. The straight line ( AB ) represents the floor, the segments [ AM ] and [ BN ] represent the legs and the seg-ment [ MN ] represents the seat of the stool. The stool’s legs are connected to the point O . The following measurements in cen-timeters are given : OA =80 ; OM =50 ; BN =143 ; ON =55 ; AB =72 Determine the length of the stool seat. Indication : it will be indicated beforehand that the seat of the stool is parallel to the ground 11. Reciprocal, Thales’ theorem, and magnitudes https://chingmath.fr chapExoCorrec/661 sacados/661 Groupe Est - Juin 2003 - 3 points ABCDEFG chapExoCorrec/3461 sacados/3461 ABCDEF chapExoCorrec/664 sacados/664 Groupe Sud - Juin 2003 - 4 points chapExoCorrec/673 sacados/673 Groupe Nord - Juin 2004 - 4 points ABCDEO chapExoCorrec/9312 sacados/9312 OABMN
ABCDEFG7km1;5km2km5km3;5km ABCDE ABCDE E.10840 Michel is taking part in a mountain bike rally on a signposted route. The route is shown in solid lines. The drawing is not to scale. Points A , B and C are aligned. The points C , D and E are aligned. The points B , D and F are aligned. The points E , F and G are aligned. The triangle BCD is right-angled at C . The triangle DEF is right-angled at E . 1 Show that the length BD is equal to 2.5 km . 2 Justify that the straight lines ( BC ) and ( EF ) are paral-lel. 3 Calculate the length DF . 4 Calculate the total length of the course. 5 Michel drives at an average speed of 16 km = h to get from point A to point B . How long will it take to go from point A to point B . Give your answer in minutes and seconds. E.10839 La régate AB = 400 ; AC = 300 ; BC = 500 ; CD = 700 the straight lines ( AE ) and ( BD ) intersect at C . The straight lines ( AB ) and ( DE ) are parallel. 1 Calculate length DE . 2 Show that the triangle ABC is right-angled. 3 Calculate the measure of the angle ABC . Round to the nearest degree. In a race, competitors must complete several laps of the course shown above. They start from point A , then pass through points B , C , D and E in that order then back through point C and then back to point A . Maltéo, the winner, took 1 h 48 min to complete all 5 of the course. The distance traveled to complete one lap is 2 880 m . 4 Calculate the total distance traveled to complete the 5 laps of the course. 5 Calculate Maltéo’s average speed. Round to the nearest whole number. 12. Theorem, reciprocal of Thales and Pythagoras E.650 The figure below is not full-scale. It is not required to be reproduced. The points A , C and E are aligned, as are the points B , C and D . The triangle ABC is rectangular in B . The following lengths are expressed in centimetres : BC = 12 ; CD = 9.6 ; DE = 4 ; CE = 10.4 1 Show that the triangle CDE is rectangular at D . 2 Deduce that the straight lines ( AB ) and ( DE ) are paral-lel. 3 Calculate the length AB . E.2680 Consider a triangle EFG such that : EF = 6 cm ; FG = 7.5 cm ; GE = 4.5 cm . 1 Construct the triangle EFG . 2 Show that the triangle EFG is right-angled and specify at which point. 3 Construct the point M middle of [ EF ] and construct the line parallel to [ EG ] passing through M ; it intersects [ FG ] at N . 4 Show that N is the middle of [ FG ] . https://chingmath.fr chapExoCorrec/10840 sacados/10840 ABCDEFG7km1;5km2km5km3;5km chapExoCorrec/10839 sacados/10839 ABCDE chapExoCorrec/650 sacados/650 Nantes - Juin 2006 ABCDE chapExoCorrec/2680 sacados/2680
MNORS A(Départ)BCDE(Départ)400m300m1000m ABCDIJMN17cm16cm ABCMN E.2168 the figure opposite is not full-scale ; we don’t ask you to reproduce it. The points N , O , R on the one hand and the points M , O , S on the other are aligned in this order. OS = 6 cm ; OM = 9 cm ; ON = 5.4 cm ; OR = 3.6 cm 1 Are the straight lines ( MN ) and ( RS ) parallel? Justify. 2 Assume SR =4.8 cm . Is the triangle ORS right-angled? Justify. 3 Using Thales’ theorem , calculate MN . 4 It will be assumed that the straight lines ( MN ) and ( NR ) are perpendicular. What is the area of the quadrilateral MNSR ? Justify. E.5047 Students participate in a foot race. Before the event, they were given a map. It is repre-sented by the figure below : It is agreed that : The straight lines ( AE ) and ( BD ) intersect at C . The straight lines ( AB ) and ( DE ) are parallel. ABC is a right triangle in A . Calculate the actual length of the path ABCDE If the work is not completed, still leave a record of the search. It will be considered in the scoring. E.10841 Consider the rectangle ABCD shown below with : AB = 17 cm ; AD = 16 cm Let I and J be the respective middles of segments [ AD ] and [ BC ] . We place the point M on the segment [ IJ ] so that the triangle AMB is isosceles at A . Finally, we note N the point of intersection of the straight lines ( AM ) and ( BC ) . Determine the measure of segment [ AN ] . 13. Thales’ theorem and trigonometry E.10764 Consider the right triangle ABC at C shown below and the points M and N belonging respec-tively to the segments [ BC ] and [ BA ] : We have the following measurements : https://chingmath.fr chapExoCorrec/2168 sacados/2168 MNORS chapExoCorrec/5047 sacados/5047 Brevet Metropole Juin 2012 A(Départ)BCDE(Départ)400m300m1000m chapExoCorrec/10841 sacados/10841 ABCDIJMN17cm16cm chapExoCorrec/10764 sacados/10764 ABCMN
ABCMN 520m200m120m600mDKJAL 133209200314OZSHX AMNUVCB BM =2.4 cm ; MC =1.2 cm ; AC =1.5 cm BN =2.6 cm ; BA =3.9 cm 1 Prove that triangle BMN is a right triangle at M . 2 a Determine the length of segment [ MN ] . b Determine the measure of angle MNB rounded to the nearest degree. E.10791 Consider the five points A , B , C , M , N such that the point B is the intersection of the straight lines ( AN ) and ( CM ) : We have the following measurements : BA =4 cm ; BC =3.2 cm ; AC =2.4 cm BM =2 cm ; BN =2.5 cm 1 Demonstrate that the triangle BMN is rectangular at M . 2 a Determine the length of the segment [ MN ] . b Determine the measure of the angle MNB rounded to the nearest degree. E.10835 The following figure, which is not to scale, shows the route Oscar must take. In the right-angled triangle DLA at L , point J belongs to segment [ DA ] and point K belongs to segment [ DL ] . Given : DL =600 cm ; KJ =200 m ; DJ =520 m ; KL =120 m 1 Show that the length DK is equal to 480 m . 2 Justify that triangle DKJ is a right triangle at K . 3 Show that segment [ DA ] measures 650 m . 4 Calculate the length of path DKJA , marked with arrows in the figure. 5 A photographer places a camera at point D . In order to film the entire race without moving the camera, angle LDA must be less than 25 o . Is this the case? 14. Contraposé of Thales’ theorem E.667 Show that the straight lines ( OZ ) and ( HS ) are not parallel: E.657 A carpenter has just created the following cabinet. The [ CB ] board rests directly on the floor. He noted the following dimensions : AC = 2 m ; AB = 2.5 m ; AM = 0.4 m CU = 0.8 m ; AN = 0.6 m ; V B = 1 m 1 Is the MN board in the horizontal position? 2 Same question for board UV . 15. Unclassified exercises E.5441 https://chingmath.fr chapExoCorrec/10791 sacados/10791 ABCMN chapExoCorrec/10835 sacados/10835 Juillet 2024 Martinique 520m200m120m600mDKJAL chapExoCorrec/667 sacados/667 133209200314OZSHX chapExoCorrec/657 sacados/657 AMNUVCB chapExoCorrec/5441 sacados/5441
ABCDEFSH ABCEJHM BFHJK4;2m7;5m9;3m12;4m10m On wants to make a teepee that will have the shape of a pyramid having as its base a rectangle ABCD of center H and as its height [ SH ] (see diagram oppo-site) . The tipi will have the following dimensions : AD = 1.60 m CD = 1.20 m SH = 2.40 m 1 Calculate the volume V of this pyramid, in m 3 . Recall that V = 1 3 × B × h où h denotes the height and B the area of the base. 2 Calculate the length BD . 3 The tepee’s frame, consisting of the rectangular ABCD frame and the four side edges coming from S , is made of bamboo rods. No written demonstration is expected in this question. Quoting a property and clearly presenting a calculation will suffice. a Show that : SD =2.60 m b A rod [ EF ] is added to the frame as shown in the draw-ing so that ( EF ) == ( AD ) and SF =1.95 m . Calculate EF . 4 We found 3 m bamboo stalks in a store. A rod can be cut to make two chopsticks, but a chopstick cannot be made by gluing together two pieces of bamboo. How many bamboo rods do you need to buy, minimum, to make the nine chopsticks for the teepee frame? E.663 Consider the triangle ABC below, such that : AB = 6 cm ; AC = 10 cm ; BC = 8 cm Let E be the point of [ AC ] such that AE =4 cm and J be the point of [ BC ] such that CJ =2.4 cm . Let H be the midpoint of [ EC ] and M the point of intersection of the lines ( BE ) and ( JH ) . (in the figure the dimensions are not respected) 1 Prove that the straight lines ( JH ) and ( AB ) are parallel. 2 Derive the calculation of the length HM . E.10837 In the configuration below, the points F , B , J and the points H , B , K are aligned : Establish that the straight lines ( HF ) and ( KJ ) are parallel. E.10838 A tourist climbs to the top of the Arc de Triomphe in Paris : Stepping back 2 m from the edge of the Arc de Triomphe, he realizes that he has aligned his vision perfectly with the edge of the monument and the foot of a lamppost standing at the same level as the monument. These eyes are estimated to be at a height of 1 m 70 from the ground. Once back down, he measures that the distance between the edge of the monument and the edge of the lamp post is 58.2 m . Determine the height of the monument, rounded to the near-est decimeter. Indication : all traces of research and writing will be taken into account during assessment. https://chingmath.fr ABCDEFSH chapExoCorrec/663 sacados/663 ABCEJHM chapExoCorrec/10837 sacados/10837 BFHJK4;2m7;5m9;3m12;4m10m chapExoCorrec/10838 sacados/10838
OABDEFC ABCDE ABCDE E.10973 In the figure below, we have : C is a circle of center O and radius 4.5 cm ; [ AB ] is a diameter of this circle and D is a point of the circle; The points B , E , A are aligned, as are the points D , F , A ; the straight lines ( BD ) and ( EF ) are parallel; BD = 5.4 cm ; DA = 7.2 cm ; AE = 2.7 cm 1 Justify that the diameter [ AB ] measures 9 cm . 2 Demonstrate that the triangle ABD is right-angled at D . 3 Calculate AF . 4 a Justify that the area of the triangle ABD is equal to 19.44 cm 2 . b Calculate the area of the disk, rounded to the hun-dredth. Hint: we’ll use : ı ≈ 3.1416 5 What percentage of the area of the disc represents the area of the triangle ABD ? E.11546 A city’s botanical garden can be represented by the quadrilateral ABCD below : We know that : AB = 500 m ; BE = 250 m ; DE = 750 m segments [ AC ] and [ BD ] intersect at point E . The figure opposite is not to scale . 1 What is the length of segment [ BD ] ? 2 Using the right triangle ABD , show that the length of segment [ AD ] , rounded to the nearest meter, is approxi-mately equal to 866 m . 3 a Calculate the sine of angle EAB . b Deduce the measure in degrees of angle EAB . 4 a Show that lines ( AB ) and ( DC ) are parallel. b Show that the length of segment [ CD ] is equal to 1 500 m . 5 A pedestrian walks around the botanical garden at an average speed of 1 ; 1 m = s . He reads on his map that the length of segment [ BC ] is approximately equal to 1 323 m . Does it take the pedestrian less than an hour to walk around the botanical garden? E.11547 Consider the triangle ACE below, where points B and D belong to segments [ AC ] and [ CE ] , respectively. The measurements are given : AB = 9 m ; BC = 6 m ; AE = 8 m ; CE = 17 m 1 Justify that triangle ACE is a right triangle at A . 2 Justify that lines ( AE ) and ( BD ) are parallel. 3 Determine the length of segment [ CD ] . https://chingmath.fr chapExoCorrec/10973 sacados/10973 Metropole 2024 OABDEFC chapExoCorrec/11546 sacados/11546 ABCDE chapExoCorrec/11547 sacados/11547 ABCDE
ABC2;5m2m1;5m ABCIJ(AB==(IJ ABCO(d1(d2 ABCI ABCOOcentre du cercle ABC EFG AEFRT E.1031 In each of the figures below, new prop-erties can be obtained. In each case, which theorem can be used to assert the existence of these new properties? a b c d e E.652 1 Consider the two half-lines [ AC ) and [ AB ) . The points represented on the half-right [ AC ) are equally distributed on it. a Name N one of the points represented on the half-line [ AC ) . b Draw the parallel to the line ( BC ) passing through N . Name M the point of intersection of this parallel with the half-line [ AB ) . c Replace the greyed-out ˇı values, present in the table below, by the measurement of the associated segment : [ AB ] [ AC ] [ BC ] [ AM ] [ AN ] [ MN ] d What can we say about the previous table? e Check whether the equality below is verified or not : AM AB = AN AC = MN BC 2 Consider the triangle EFG below and wish to construct a ˇ agrandissement ı and verify some properties of these two triangles : a Place the point P on the half-right [ EF ) and the point Q on the half-right [ EG ) such that : EP = 2.5 × EF ; EQ = 2.5 × EG . b What can be said about the straight lines ( FG ) and ( PQ ) ? c Compare the measurements of [ FG ] and [ PQ ] . E.8934 Consider the figure opposite, made freehand and not to scale. The following information is given : The straight lines ( ER ) and ( FT ) intersect at A . AE =8 cm , AF =10 cm , EF =6 cm AR =12 cm , AT =14 cm 1 Show that the triangle AEF is right-angled E . 2 Are the lines ( EF ) and ( RT ) parallel? https://chingmath.fr chapExoCorrec/1031 sacados/1031 ABC2;5m2m1;5m ABCIJ(AB==(IJ ABCO(d1(d2 ABCI ABCOOcentre du cercle chapExoCorrec/652 sacados/652 ABC EFG chapExoCorrec/8934 sacados/8934 AEFRT