Grade 8 / Theorem of Thales 44 exercises (including 43 corrected)

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xyOCAPDQ ABCMN DEFPQ ChingQuizz : 9 exercises available for Quizz assessment : 1. Introductory activity E.1142 Consider the two half-lines [ Ox ) and [ Oy ) below : 1 Using a compass, check that the points and crosses are evenly distributed on each of these half-lines. 2 a Among the points placed on [ Oy ) , name B the only point on the line ( Oy ) such that ( CD ) == ( AB ) . b Justify that : OC OA = 3 7 . c Justify that the two ratios OC OA and OD OB are equal. 3 a Is the line ( PQ ) parallel to the line ( CD ) ? b Compare the ratios OC OP and OD OQ . What do we notice? E.1138 1 In the triangle ABC shown below, the points M and N belong to the segments [ AB ] and [ AC ] respectively: a Do the straight lines ( BC ) and ( MN ) appear parallel or secant? b Taking measurements on the triangle ABC below, complete the table below : AB AC BC Triangle n ◦ 1 AM AN MN Triangle n ◦ 2 Quotient des longueurs 2 In the triangle DEF shown below, the points P and Q belong to the segments [ DE ] and [ DF ] respectively: a Do the straight lines ( EF ) and ( PQ ) appear parallel or secant? b Taking measurements on the triangle DEF below, complete the table below : DE DF EF Triangle n ◦ 1 DP DQ PQ Triangle n ◦ 2 Quotient des longueurs 3 Complete the sentence below as appropriate : In a triangle ABC such that M ∈ [ AB ] and N ∈ [ AC ] , if the straight lines ( MN ) and ( BC ) are parallel, then the mea-sures of one triangle are . . . . . . . . . . . . . . . . . . . the measures of the other. We then have the equality of the following ratios : . . . . . . . . . . . . = . . . . . . . . . . . . = . . . . . . . . . . . . https://chingmath.fr chapExoCorrec/1142 sacados/1142 xyOCAPDQ chapExoCorrec/1138 sacados/1138 ABCMN DEFPQ
GHJIKVWXYZ ABCMNABCMN2cmABCMNABCMN2cm6cmABCMNABCMN2cmABCMNABCMN2cm6cm3cm JesaisJ’utiliseJ’endéduisLes pointsA,M,Bsont alignés.Les pointsA,N,Csont alignés.(MN==(BCD’aprèslethéorèmedeThalès,onal’égalité des quotients:AMABANACMNBCChaînonsdéductifs 2. Cross product E.8922 Proposition: (of the cross product) Let there be four numbers a , b , c , d four numbers with b =0 and d =0 , If we have equality of ratios a b = c d then we have equality of products a × d = b × c . We note : a b = c d = ⇒ a × d = b × c For each question, give the decimal writing of the number x verifying the equality: a 2 5 = x 4 b 3 x = 12 5 c x 14 = 5 8 d 5 6 = x 24 E.2163 In each case, find the length AB veri-fying the equality and give its value in the form of a simplified fraction : a AB 3 = 5 9 b 2 AB = 7 3 c 5 8 = 8 AB d AB 3 = 8 9 3. Introduction to the Thales theorem E.8913 Thales’ theorem: If the points A , B , M are aligned, the points A , C , N are aligned and if the straight lines ( BC ) and ( MN ) are parallel then : AM AB = AN AC = MN BC Note: the equality of length quotients reflects the propor-tionality of lengths between nested ˇı triangles when two of their sides are parallel. We have represented two Thales configurations où ( GH ) == ( IJ ) and ( XY ) == ( V W ) . For each of these triangles, write the equality of quotients obtained by using Thales’ theorem. E.8917 Commented example: Consider the triangle ABC shown below : The points M and N belong respectively to the segment [ AB ] and [ AC ] and are such that the straight lines ( MN ) and ( BC ) are parallel. We have the numerical application : 2 6 = 3 AC = MN BC Nous will use the equality: 2 6 = 3 AC D From 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 to the segmets [ TZ ] and [ TX ] respectively such that ( IL ) == ( XZ ) . https://chingmath.fr chapExoCorrec/8922 sacados/8922 chapExoCorrec/2163 sacados/2163 chapExoCorrec/8913 sacados/8913 GHJIKVWXYZ chapExoCorrec/8917 sacados/8917 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 ???? SHABO ACEBDACEBD4;2cmACEBDACEBD4;2cm14cmACEBDACEBD4;2cmACEBDACEBD4;2cm14cm12cm DEFRS2;1cm1;8cm4;2cm 1 Complete the deductive chain below : 2 Use the equality of quotients from question 2 to deter-mine the measure of segment [ TX ] . E.8937 Thalès of Milet lived in Asia Minor from around 625 to 547 BC. According to tradition, he was the first Greek math-ematician. A theorem in the theory of triangles is named after him. He is said to have applied geometrical principles to the measurement of pyramid heights and distances at sea Universalis.fr To measure the height of the Khéops pyramid, legend has it that the first form of Thales of Miletus’ theorem would have been : ˇ the ratio of the height of my stick to that of the pyra-mid is equal to the ratio of the shadow cast by my stick to that cast by the pyramid ı. By moving the stick and planting it ˇ right ı, here’s the dia-gram (not shown in full size) he would have used to measure the pyramid: The segment [ SH ] represents the height of the pyramid, [ AB ] the stick planted straight, [ HO ] the shadow cast by the pyra-mid and [ AO ] that of the stick. We give the following measurements : OH = 300 m ; OA = 3.1 m ; AB = 1.5 m Like Thales of Millet, determine the height of the Cheops pyramid rounded to the nearest metre. 4. Theorem of Thales E.8959 In the triangle ACE , the line ( BD ) is parallel to ( CE ) . Determine the measure of the segment [ BD ] . E.8958 In the triangle DEF , the straight lines ( EF ) and ( RS ) are parallel to each other. Determine the measure of seg-ment [ ER ] . https://chingmath.fr 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 chapExoCorrec/8937 sacados/8937 ???? SHABO chapExoCorrec/8959 sacados/8959 ACEBDACEBD4;2cmACEBDACEBD4;2cm14cmACEBDACEBD4;2cmACEBDACEBD4;2cm14cm12cm chapExoCorrec/8958 sacados/8958 DEFRS2;1cm1;8cm4;2cm
TZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cmTZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cm5cm ABCMN2;4cm3cm3;6cm ACBNMACBNM10cm2;6cm6;5cm TMNKLTMNKL5cmTMNKLTMNKL5cm7;5cmTMNKLTMNKL5cmTMNKLTMNKL5cm7;5cm6;5cm E.8960 In the triangle TXZ , the lines ( IL ) and ( XZ ) are parallel to each other. Determine the measure of the segment [ TX ] . E.4768 In the triangle ABC , the straight lines ( MN ) and ( BC ) are parallel to each other. Determine the measure of segment [ MB ] . E.1140 In the triangle ABC , the lines ( MN ) and ( BC ) are parallel to each other. Determine the measure of the segment [ AN ] . E.1135 In the triangle TMN , the line ( KL ) is parallel to ( MN ) . Determine the measure of the segment [ TN ] . E.662 1 Construct the triangle ABC such that : AB =7.5 cm ; BC =10 cm ; AC =12.5 cm . 2 Show that the triangle ABC is rectangular. 3 a M is a point on segment [ BC ] such that BM =4 cm . Place the point M and construct the line ( d ) parallel to the line ( AC ) through M . The line ( d ) intersects [ AB ] at N . b Calculate BN and MN . E.1903 1 a Draw a rectangle ABCD such that : AB =1.5 cm ; AD =6 cm b Place the point M on [ AB ) such that AM =6 cm . c The line ( DM ) intercepts the segment [ BC ] at I . 2 Determine the length of segment [ BI ] . 3 Determine the length of segment [ MI ] , rounded to the nearest centimeter. E.11639 In the triangle TXZ , the lines ( IL ) and ( XZ ) are parallel to each other. Determine the measure of the segment [ TX ] . 5. Thales’ theorem and fractions https://chingmath.fr chapExoCorrec/8960 sacados/8960 TZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cmTZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cm5cm chapExoCorrec/4768 sacados/4768 ABCMN2;4cm3cm3;6cm chapExoCorrec/1140 sacados/1140 ACBNMACBNM10cm2;6cm6;5cm chapExoCorrec/1135 sacados/1135 TMNKLTMNKL5cmTMNKLTMNKL5cm7;5cmTMNKLTMNKL5cmTMNKLTMNKL5cm7;5cm6;5cm chapExoCorrec/662 sacados/662 chapExoCorrec/1903 sacados/1903 sacados/11639 TZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cmTZXLITZXLI5;5cmTZXLITZXLI5;5cm9;9cm5cm
AMNBC655245DRSEF9243125 AMNBC523412DRSEF3273133 AOKSR ABCDO E.4770 1 In the triangle ABC , the lines ( MN ) and ( BC ) are paral-lel to each other. Determine the measure of the segment [ AC ] . 2 In the triangle DEF , the lines ( RS ) and ( EF ) are paral-lel to each other. Determine the measure of the segment [ EF ] . E.4771 1 In the triangle ABC , the lines ( MN ) and ( BC ) are paral-lel to each other. Determine the measure of the segment [ AN ] . 2 In the triangle DEF , the lines ( RS ) and ( EF ) are paral-lel to each other. Determine the measure of the segment [ DR ] . 6. Problems around the Thales theorem E.4330 In the configuration opposite, the straight lines ( SA ) and ( OK ) are parallel. We know that : SA = 5 cm ; OA = 3.8 cm OR = 6.84 cm ; KR = 7.2 cm The questions for this exercise have been deleted, but below remain calculations made by a student, in response to the missing questions : 1 6.84 − 3.8 = 3.04 2 5 × 6.84 3.04 = 11.25 3 7.2 + 6.84 + 11.25 = 25.29 Using all the previous calculations, write down the questions the student has answered, and write out their answers pre-cisely. E.2190 Consider the figure below, which is not drawn to full size. The unit of length is centimeters. The straight lines ( CD ) and ( OA ) are perpendicular. We give : OA = 9 ; OB = 12 ; AB = 15 ; AC = 3 1 Demonstrate that the triangle AOB is right-angled and deduce that the straight lines ( CD ) and ( OB ) are paral-lel. 2 Justifying the reasoning, show that : CD =4 . 3 A student states that the area of the triangle AOB is equal to three times the area of the triangle ACD . What do you think of this statement? Justify your an-swer. https://chingmath.fr chapExoCorrec/4770 sacados/4770 AMNBC655245DRSEF9243125 chapExoCorrec/4771 sacados/4771 AMNBC523412DRSEF3273133 chapExoCorrec/4330 sacados/4330 Amerique du Nord Juin 2011 AOKSR chapExoCorrec/2190 sacados/2190 Groupe est - Septembre 2006 - 5 points ABCDO
2;7m497;3m1;75m ABCDIJKLMN ABCDMNP4;5cm2;7cm3;6cm6;8cm ABCMNPD E.1902 A man measuring 1.75 m standing upright in the vicinity of the Eiffel Tower positions himself so that the shadow passes just over his head. His shadow falls at 2.7 m from him and this is 500m from the center of the Eiffel Tower. How high is the Eiffel Tower? (rounded to the nearest meter) E.4756 Let a be a non-zero positive num-ber. Consider the rectangle ABCD of length 3 a and width a ; divide this rectangle into three squares of equal measure : 1 Prove the equality: IM = KN . 2 How can the rectangle ABCD be divided into nine rect-angles that are all the same? 7. Double application of Thales’ theorem E.4795 Consider the configuration below où the lines ( BD ) and ( PM ) are parallel. 1 Show that : AC =6 cm . 2 Determine the measure of segment [ AM ] . E.1137 Let A , M , N , P be four non-aligned points of the plane such that : The points B , C and D belong to the segments [ AM ] , [ AN ] , [ AP ] respectively. The straight lines ( BC ) and ( MN ) are parallel, as are the straight lines ( CD ) and ( NP ) . 1 Justify the equality of the two ratios AB AM and CD NP . 2 The following measurements are given : AM =3 cm ; AB =1.2 cm ; PN =1.8 cm Deduce from the previous question the measure of seg-ment [ CD ] . https://chingmath.fr chapExoCorrec/1902 sacados/1902 2;7m497;3m1;75m chapExoCorrec/4756 sacados/4756 ABCDIJKLMN chapExoCorrec/4795 sacados/4795 ABCDMNP4;5cm2;7cm3;6cm6;8cm chapExoCorrec/1137 sacados/1137 ABCMNPD
ABCDMNO3;6m4;8m1;5m ABCMN4;8cm5;2cm3cm 384cmABC:poutresenboisdediamètre100mm:barresdemaintienlatéralesenboisABCMNH290cm342cmABCestuntriangleisocèleenA.HestlemilieudeBC.(MNestparallèleà(BC.Vue d’ensembleVue de côté E.3503 In the configuration below, the points M , N , O belong to the segments [ AB ] , [ AC ] , [ AD ] , respectively; moreover, we have : ( BC ) == ( MN ) ; ( CD ) == ( NO ) Determine the length of segment [ CD ] . 8. Thales’ theorem and other theorems E.3452 In the plane, we consider the config-uration below : The properties of the figure are as follows : The point C belongs to the line [ AN ] ; point B belongs to the line [ AM ] ; triangle ABC is a right triangle in B ; lines ( BC ) and ( MN ) are parallel. 1 Determine the measure of segment [ BC ] . 2 a Determine the length of segment [ AN ] . b Give the length of segment [ AM ] . E.8931 A company manufactures gantries to install swings on playgrounds. Handout 1: sketch of a gantry Paper 2: material cost Poutres in wood of diame-ter 100 mm : Length 4 m : 12.99 e the unit Length 3.5 m : 11.75 e per unit Length 3 m : 10.25 e per unit Wooden side rails: Length 3 m : 6.99 e each Length 2 m : 4.75 e per unit Length 1.5 m : 3.89 e the unit Set of fasteners needed for one gantry: 80 e . Set of two swings for a portico: 50 e . 1 Determine the height AH of the gantry, rounded to the nearest cm . 2 The hold-down bars must be attached at 165 cm from the top ( AN =165 cm ) . Show that the length MN of each holding bar is approximately 140 cm . https://chingmath.fr chapExoCorrec/3503 sacados/3503 ABCDMNO3;6m4;8m1;5m chapExoCorrec/3452 sacados/3452 ABCMN4;8cm5;2cm3cm chapExoCorrec/8931 sacados/8931 384cmABC:poutresenboisdediamètre100mm:barresdemaintienlatéralesenboisABCMNH290cm342cmABCestuntriangleisocèleenA.HestlemilieudeBC.(MNestparallèleà(BC.Vue d’ensembleVue de côté
ABCDEFGHS ABCDMN5;6cm4;2cm ABCDEFGHIJRugbyRugbyFoot312m29m72m52m48m288m ABCDPQRS13;5cm31;5cm8;4cm19;6cm 3 Show that the minimum cost of such a gantry equipped with swings is 196.98 e . 4 The company wants to sell this equipped swing set 20 % for more than its minimum cost. Determine this selling price rounded to the nearest hundredth. E.2170 SABCD is a pyramid with rectangular base ABCD , height [ SA ] . We give : SA = 15 cm ; AB = 8 cm ; BC = 11 cm 1 Calculate the volume V 1 of the pyramid SABCD . 2 Show that : SB =17 cm . 3 Note E the point of [ SA ] such that SE =12 cm and F the point of [ SB ] such that SF =13.6 cm . Show that the lines ( EF ) and ( AB ) are parallel. E.4755 Consider the rectangle ABCD of length 5.6 cm and width 4.2 cm . The point M belongs to the segment [ AC ] such that AM = 4 cm . The line perpendicular to the line ( AB ) through the point M intercepts the segment ( AB ) at the point N . 1 Justify that the lines ( MN ) and ( BC ) are parallel. 2 Determine the value of length AC . 3 Determine the value of length AN . E.5690 In this exercise, if the work is not completed, still leave a trace of the research. It will be taken into account in the assessment. BONVIVRE has a playground bordered by a bicycle path. The bike path is shaped like a rectangle ABCD with ˇ three of the corners ı removed. The path from G to H is a circular arc; the paths from E to F and from I to J are segments. The straight lines ( EF ) and ( AC ) are parallel. What is the length of the cycle path? Justify your answer. Hint: we’ll use : ı ≈ 3.1416 lengths should be rounded to the nearest centimetre. E.4769 Consider a rectangle ABCD of length 31.5 cm and width 19.6 cm . Let P be the point in [ AB ] such that AP =13.5 cm . Let Q be the point of intersection of the line parallel to the line ( DB ) passing through the point P and with the line ( AD ) . Let S be the point of [ BC ] such that CS =8.4 cm . We de-note R the point of intersection of the line parallel to the line ( DB ) through the point S and with the line ( DC ) . 1 Determine the measure of segment [ BD ] . 2 Show that the quadrilateral PQRS is a parallelogram. https://chingmath.fr chapExoCorrec/2170 sacados/2170 ABCDEFGHS chapExoCorrec/4755 sacados/4755 ABCDMN5;6cm4;2cm chapExoCorrec/5690 sacados/5690 Brevet - Asie Juin 2013 ABCDEFGHIJRugbyRugbyFoot312m29m72m52m48m288m chapExoCorrec/4769 sacados/4769 ABCDPQRS13;5cm31;5cm8;4cm19;6cm
ABCDEFM36 ABCDEFGH5cm9cm3cm MGPUA4;5m3;5m5;6m7;2m JesaisJ’utiliseJ’endéduisLes pointsM,A,Pet les pointsM,U,Gsont alignés dans le même ordre.MAMPMUMGD’aprèslaréciproqueduthéorèmedeThalès:(AU==(PGChaînonsdéductifs E.5923 In this exercise, any trace of research, however incomplete, will be taken into account in the assessment. Consider the figure below, which is not full-scale. BCDE is a square with 6 cm sides. The points A , B and C are aligned and AB =3 cm . F is a point on the seg-ment [ CD ] . The straight line ( AF ) intersects the segment [ BE ] at M . Determine length CF by calculation or construction so that lengths BM and FD are equal. E.9069 An ant is walking on the straight path ABCDEFGH shown below : It chooses the shortest path to go from point G to point A . Determine the length of this path. Note: round to the nearest tenth of a millimeter, if neces-sary. 9. Reciprocal of Thales’ theorem (decimal quotient) E.3470 Prove, without the use of a calcula-tor, that each of the pairs of quotients below are equal: a 1.4 6 et 7 30 b 3 1.2 et 5 2 c 0.01 0.04 et 0.3 1.2 d 7.2 1.6 et 5.4 1.2 E.8928 Exemple commenté: Consider the triangle MGP shown below and the points A and U belonging to the segments [ MP ] and [ MG ] re-spectively: 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 the triangle ANT and the two points R and C be-longing to the segments [ AN ] and [ AT ] respectively. We have the measurements : AN =7.5 cm ; AR =4.8 cm ; AC =3.2 cm ; AT =5 cm https://chingmath.fr chapExoCorrec/5923 sacados/5923 ABCDEFM36 chapExoCorrec/9069 sacados/9069 ABCDEFGH5cm9cm3cm chapExoCorrec/3470 sacados/3470 chapExoCorrec/8928 sacados/8928 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 0;75m3m2;5m2mPQRST 6m2;4m2;4m1;6mABCDE 8;4m3;6m7m3mPQRST 1 Prove that equality of quotients : AR AN = AC AT 2 Complete the deductive chain below : E.5668 Consider the triangle PST shown below and the two points Q and R belonging to the segments [ PS ] and [ PT ] respectively. We have the following measurements : PR = 2 m ; PT = 2.5 m ; PQ = 3 m ; QS = 0.75 m Show that the straight lines ( QR ) and ( ST ) are parallel. E.8929 Consider the two configurations below composed of two triangles ADE and PST . Consider the triangle ADE shown opposite and the two points B and C belonging to the segments [ AD ] and [ AE ] respectively. We have the measurements : AB =1.6 m ; BD =2.4 m AC =2.4 m ; AE =6 m Show that the straight lines ( BC ) and ( DE ) are parallel. 10. Reciprocal of Thales’ theorem (rational quotient) E.8930 Proposition: (reciprocal of the cross product) Let a , b , c , d be four non-zero numbers. Si a × d = b × c alors a b = c d For each question, justify that the numbers A and B are equal: a A = 2.4 5.6 et B = 3.6 8.4 b A = 2.2 3.4 et B = 3.3 5.1 E.8936 Consider the triangle PST shown below and the two points Q and R belonging to the segments [ PS ] and [ PT ] respectively. We have the following measurements : PR = 3 m ; PT = 7 m ; PQ = 3 ; 6 m ; PS = 8 ; 4 m https://chingmath.fr 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 chapExoCorrec/5668 sacados/5668 0;75m3m2;5m2mPQRST chapExoCorrec/8929 sacados/8929 6m2;4m2;4m1;6mABCDE chapExoCorrec/8930 sacados/8930 chapExoCorrec/8936 sacados/8936 8;4m3;6m7m3mPQRST
AEFRT GTESFR9cm2cm4cm1;5cm BACMN12m20m10m9m ABCDE Show that the lines ( QR ) and ( ST ) are parallel. E.8935 We consider the figure opposite, made freehand and which is not to scale. The following information is given : The straight lines ( ER ) and ( FT ) intersect at A . AE =4.2 cm , AF =2.7 cm , EF =6 cm AR =9.8 cm , FT =3.6 cm Are the lines ( EF ) and ( RT ) parallel? Justify your answer. 11. Reciprocal and Thales’ theorem E.665 In the figure opposite, the straight lines ( SF ) and ( TE ) are parallel. The points R , S and T are aligned in this order. Points R , F , E and G are aligned in this order. SR =2 cm ; ST =4 cm ; RF =1.5 cm ; EG =9 cm 1 Demonstrate that : RE =4.5 cm . 2 Are the straight lines ( ES ) and ( TG ) parallel? Justify. 12. Thales’ Reciprocal and other theorems E.8961 Consider the triangle ABC shown below where the points M and N belong to the segments [ BA ] and [ BC ] respectively, where the triangle BMN is right-angled at M and the following distances are known : BM = 12 m ; BA = 20 m ; NC = 10 m ; MN = 9 m 1 Determine the measure of segment [ BN ] . 2 Show that the lines ( MN ) and ( AC ) are parallel. 13. Unclassified exercises E.651 In the triangle CDE , A is a point on the segment [ CE ] ; B is a point on the segment [ CD ] . In the diagram opposite, the lengths shown are not exact. The following lengths are given : AC = 8 cm ; CE = 20 cm ; BC = 6 cm CD = 15 cm ; DE = 25 cm https://chingmath.fr chapExoCorrec/8935 sacados/8935 AEFRT chapExoCorrec/665 sacados/665 Groupe Ouest - Septembre 2002 - 4 points GTESFR9cm2cm4cm1;5cm chapExoCorrec/8961 sacados/8961 BACMN12m20m10m9m chapExoCorrec/651 sacados/651 Groupe Nord - Juin 2004 - 4 points ABCDE
1 Show that the straight lines ( AB ) and ( DE ) are parallel. 2 Is the triangle CDE right-angled? Justify. 3 Calculate AB 4 Calculate the value, rounded to the nearest degree, of the angle CDE E.668 ABCD is a parallelogram such that : AB = 8 cm ; AD = 4.5 cm E is a point on the line ( AD ) such that : AE = 1.5 cm and E is not on segment [ AD ] . The straight line ( EC ) intersects the segment [ AB ] at M . 1 Calculate AM . 2 Place the point N on the segment [ DC ] such that : DN = 3 4 × DC 3 Demonstrate that the straight lines ( AN ) and ( EC ) are parallel. https://chingmath.fr chapExoCorrec/668 sacados/668 ?? - Juin 2000 - 6 points