Grade 9 / Other 38 exercises (including 17 corrected)

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1. The Pythagorean triplets E.1959 Study of the triplet m 2 − n 2 ; 2 mn ; m 2 + n 2 : 1 Show that, for n and n positive integer, the following triplet:is a Pythagorean triplet: m 2 − n 2 ; 2 mn ; m 2 + n 2 Study of a special case: 2 a Verify that this triplet is Pythagorean. By identification with the general triplet, we must have the two integers m and n that simultaneously verify the three lines (the following system) : m 2 − n 2 = 15 2 mn = 8 m 2 + n 2 = 17 b Solve this system : that is, find the (unique) values of m and n verifying these three lines. For all Pythagorean triplets: 3 For any Pythagorean triplet a ; b ; c , we admit that the fractions c + a 2 and c − a 2 are perfect squares. Let’s say: m 2 = c + a 2 ; n 2 = c − a 2 Show that m and n thus chosen satisfy the question. (for the verification of b we’ll just show that b 2 =4 m 2 n 2 ) E.1960 Triplet study 2 mn ; m 2 − n 2 ; m 2 + n 2 : 1 We will establish that for any positive integer m and n , the triplet 2 mn ; m 2 − n 2 ; m 2 + n 2 is a Pythagorean triplet: a Expand and reduce : m 2 − n 2 m 2 − n 2 b Establish the following equality: 2 mn 2 + m 2 − n 2 2 = m 2 + n 2 2 Study of a special case: 2 Study of the triplet 16 ; 30 ; 34 a Verify that this triplet is Pythagorean. By indentification with the general triplet, we must have the two integers m and n simultaneously verifying the three lines (the following system) : m 2 − n 2 = 16 2 mn = 30 m 2 + n 2 = 34 b Solve this system. For all Pythagorean triplets: 3 For any Pythagorean triplet a ; b ; c , we admit that the fractions c + a 2 and c − a 2 are perfect squares. Let’s posit : m 2 = c + a 2 ; n 2 = c − a 2 a Establish the following equalities: a = m 2 − n 2 ; c = m 2 + n 2 b Expand and reduce the following product : ( c + a )( c − a ) c Show that : b 2 = 4 m 2 n 2 We have just established : b =2 mn E.1961 Study of the triplet 2 mn ; m 2 − n 2 ; m 2 + n 2 : 1 We will establish that for any positive integer m and n , the triplet 2 mn ; m 2 − n 2 ; m 2 + n 2 is a Pythagorean triplet: a Expand and reduce to obtain the following equality: m 2 + n 2 m 2 + n 2 = m 4 + 2 m 2 n 2 + n 4 b Expand and reduce : m 2 − n 2 m 2 − n 2 c Simplify writing 2 mn 2 d Establish the following equality: 2 mn 2 + m 2 − n 2 2 = m 2 + n 2 2 Study of a special case: 2 Study of the triplet 16 ; 30 ; 34 a Verify that this triplet is Pythagorean. By identification with the general triplet, we must have the two integers m and n that simultaneously verify the three lines (the following system) : m 2 − n 2 = 16 2 m · n = 30 m 2 + n 2 = 34 b Focusing on the line 2 m · n =30 , determine the value of the integers m and n . For all Pythagorean triplets: 3 For any Pythagorean triplet a ; b ; c , we admit that the fractions c + a 2 and c − a 2 are perfect squares. Let’s posit : m 2 = c + a 2 ; n 2 = c − a 2 a Establish the following equalities: a = m 2 − n 2 ; c = m 2 + n 2 b Expand and reduce the following product : ( c + a )( c − a ) c Using the fact that a ; b ; c is a Pythagorean triplet, show that : b 2 = 4 m 2 n 2 We have just established : b =2 mn https://chingmath.fr sacados/1959 sacados/1960 sacados/1961
6m4m2;5m 6cm3cm2cmABCDSM E.1954 1 Let’s show that the triplet 2 mn ; m 2 − n 2 ; m 2 + n 2 is a Pythagorean triplet: a Expand and reduce : m 2 − n 2 m 2 − n 2 b Expand and reduce : m 2 + n 2 m 2 + n 2 c Simplify writing 2 mn 2 d Establish the following equality: 2 mn 2 + m 2 − n 2 2 = m 2 + n 2 2 Now we’ll show that any Pythagorean triplet can be written as 2 mn ; m 2 − n 2 ; m 2 + n 2 . 2 Study of the triplet 16 ; 30 ; 34 a Verify that this triplet is Pythagorean. By identification with the general triplet, we must have m and n simultaneously verifying the three lines (the fol-lowing system) : m 2 − n 2 = 16 2 mn = 30 m 2 + n 2 = 34 a Solve the previous system. 2 For a Pythagorean triplet a ; b ; c , let’s posit : m 2 = c + a 2 ; n 2 = c − a 2 a Establish the following equalities: a = m 2 − n 2 ; b =2 mn ; c = m 2 − n 2 2. Open problems E.5790 A goat is in a pasture of rectangular shape and size 6 m × 4 m . It is tied to a shed, square in shape with sides measuring 2.5 m , as shown below. What is the minimum length of rope needed for the goat to walk the entire lawn? E.5791 Two plane trees, the first measuring 17 m in height and the second measuring 10 m in height, are sepa-rated by 24 m . On top of each of the trees is a bird that will fly in a straight line to a fountain whose top is set 1 m high. This fountain is located in the middle between the two trees. From the tall tree, the first bird launches itself toward the fountain at a speed of 50 km = h . What must be the minimum speed of the second bird in order to arrive at the fountain first? E.5792 A square with 2 m side is shown hatched in the representation below. It lies inside the rectangle ABCD , of dimensions 6 m × 3 m and rests on the side [ AB ] : What must be the position of the point M so that the square lies within the triangular ABC ? E.5793 The traditional dwelling of the North American Plains Indians is the tepee. A tipi consists of long wooden rods leaning against each other, an outer shell made of animal skins and a door that always faces east. Each wooden pole measures 21 feet and protrudes 3 feet. The radius of the cir-cle traced on the ground measures 7.5 feet. The great Indian chief wants to cap the circle formed by the top of his tepee poles with a feather hat. What should be the diameter of his hat? Explain your approach in a text presenting your calculations and arguments, and illustrate with a figure. 3. Decomposition into prime factors E.636 1 write the following numbers as an irreducible fraction : A = 1 − 7 12 × 4 5 ; B = 42 7 ÷ 12 35 https://chingmath.fr sacados/1954 A finir sacados/5790 6m4m2;5m sacados/5791 sacados/5792 6cm3cm2cmABCDSM sacados/5793 chapExoCorrec/636 sacados/636
KadoLapatAngazPirasMegalNuben AngazKadoLapatMegalNubenPirasAngazKadoLapatMegalNubenPiras5505003005003003008508505501000800450600250 Ligne BLigne ALigne CD’iciÀ Là Représente une station sur une des lignes de métroReprésenteunejonction,c’est-à-direunestationoùexisteunecorres-pondancepermettantdechangerdelignedemétro(LignesA,BouC) 2 Simplify the following fraction to make it irreducible : C = 3 8 × 5 8 × 7 3 15 8 × 14 2 E.638 Perform the following calculations and give the result as irreducible fractions : A 1 3 + 1 2 × 1 − 1 3 b 24 5 ÷ 12 30 c 1 4 + 1 2 3 + 1 6 d 3 2 × 11 − 4 × 4 3 2 6 × 11 2 E.3527 Write the following numbers in the form 2 n × 3 m × 5 k où the integers n , m , k relative integers. A 18 × 15 2 × 12 4 b 6 10 × 5 3 × 10 2 15 7 × 2 3 c ( − 3) 3 × 15 2 × ( − 4) 3 16 2 × ( − 9) 2 4. Problems with the PISA survey E.5794 Below is a map of the roads in a region: Below are given the shortest road distances between cities, expressed in kilometers: Calculate the shortest distance by road between Nuben and Kado E.5795 The diagram below shows a section of a Zedonian city’s public transport network, including three metro lines. The price depends on the number of stations crossed (not in-cluding the departure station) . The cost amounts to 1 zed per station. Travel time between two sucessive stations is approximately 2 minutes. The time required to change lines at a junction is approxi-mately 5 minutes. On the diagram, we can see the station o where a traveler is at the moment (ˇFrom hereı) and o where he wishes to go (ˇA làı) . Indicate on the diagram the best route (in terms of du-ration and cost) and give the price you will pay, as well as the approximate duration of the journey. https://chingmath.fr chapExoCorrec/638 sacados/638 chapExoCorrec/3527 sacados/3527 sacados/5794 KadoLapatAngazPirasMegalNuben AngazKadoLapatMegalNubenPirasAngazKadoLapatMegalNubenPiras5505003005003003008508505501000800450600250 sacados/5795 Ligne BLigne ALigne CD’iciÀ Là Représente une station sur une des lignes de métroReprésenteunejonction,c’est-à-direunestationoùexisteunecorres-pondancepermettantdechangerdelignedemétro(LignesA,BouC)
AEBFCGDHEntréeSortie x-20246y-3-2-11ABCDE -4-3-2-1234I-2-1234JOCDAB E.5796 The system below represents a canal sys-tem designed to irrigate cultivated plots. Gates A to H can be opened or closed to bring water whereù is needed. When a valve is closed, water does not pass through. In this problem, the aim is to identify a valve that is blocked, preventing water from flowing through the channel system. Michel noticed that the water wasn’t always flowing where it was supposed toù. He thinks one of the valves is stuck in the closed position, so it won’t open, even when commanded to open. Michel realizes that, when the valves are set as shown in the table below, there is no water flowing out of the outlet, indi-cating that at least one of the valves set to ˇ open position ı is actually locked in the closed position. A B C D E F G H Ouverte Fermée Ouverte Ouverte Fermée Ouverte Fermée Ouverte For each of the failures described below, indicate whether wa-ter will flow to the outlet. Circle ˇ Yes ı or ˇ Non ı for each fault. Panne L’eau s’écoulera t-elle jusqu’à la sortie? The A valve is locked in the closed po-sition. All others operate correctly ac-cording to the settings in the table. Yes / No Valve D is locked in closed position. All others operate correctly according to the settings in the table. Yes / No Valve F is locked in closed position. All others operate correctly according to the settings in the table. Yes / No 5. Lines and affine functions E.2610 Consider the plane provided with an orthonormal coordinate system ( O ; I ; J ) and a straight line shown below : 1 Determine the coordinates of the points : A ; B ; C ; D ; E 2 a Without justification, give the nature of the trian-gles ADE and ABC . b In these two triangles, determine the following trigono-metric values : tan DAE ; tan CAB c What can we say about the points A , B , C ? 3 Using the coordinates of the points, determine the value of the following two quotients : y A − y C x A − x C ; y A − y D x A − x D 4 Verify that the coordinates of the points A , D , C verify the following equality: y = 0.5 x − 2 E.965 In the reference frame O ; I ; J orthonor-mal, consider the points A , B , C , D shown below : 1 Give the coordinates of the points A , B , C and D . 2 a Draw the line ( AB ) . Determine the value of the intercept of the line ( AB ) . b Place the point M (0 ; − 1) . Determine the value of the trigonometric ratio: tan ∠ MAB c Derive the expression for the linear function f that ad-mits the line ( AB ) as its representation. 3 a Draw the line ( CD ) . Determine the value of the intercept of the line ( CD ) . b Place the point N (2 ; 2.5) . Determine the value of the trigonometric ratio: tan ∠ NCD c Let g be the linear functionthat admits as its represen-tation the line ( CD ) . Which of the following is the expression for the function g ? g ( x ) = 0.5 x + 2.5 ; g ( x ) = − 0.5 x + 2.5 https://chingmath.fr sacados/5796 AEBFCGDHEntréeSortie chapExoCorrec/2610 sacados/2610 x-20246y-3-2-11ABCDE chapExoCorrec/965 sacados/965 -4-3-2-1234I-2-1234JOCDAB
ABCDF4x6 x012345678y2468101214161820222426 E.2483 A sports club offers its customers three types of rates : Tariff 1: payment of 1 000 F for each session. Tariff 2: payment of a monthly card of 4 000 F plus 500 F per session attended. Tariff 3: a monthly subscription of 11 500 F 1 Mr Bob Iscotto plans to attend 10 sessions per month. Calculate his expenditure with each of the rates. 2 Monsieur Ray Gimesseq does not know how many ses-sions he will attend in a month. We call x the number of sessions attended in the month. a Express as a function of x , the prices p 1 , p 2 , p 3 to be paid in each case. b On graph paper, construct a landmark where : 1 cm represents 2 sessions in abscissa ; 1 cm represents 1 000 F on the ordinate ; the x-axis and y-axis are perpendicular. Draw the graphical representations of the functions t 1 and t 2 such that : t 1 ( x ) = 1 000 x ; t 2 ( x ) = 500 x + 4 000 . 3 a Solve the system : y = 1 000 x y = 500 x + 4 000 b Copy and complete the following sentence : ˇ Graphically, the solution of this system corresponds to the location où. . . ı c From how many sessions is rate 3 more advantageous than rate 2? 4 Copy and complete the following sentences : a From zero to . . . séances, MI Ray Gimesseq should choose rate . . . . b From . . . à . . . séances, Mr. Ray Gimesseq should choose rate . . . . c From . . . séances, Mr. Ray Gimesseq should choose tariff . . . . Mr. Ray Gimesseq thanks you for your advice! Note : In New Caledonia, the Pacific franc is used. For in-formation 100 Pacific francs are worth approximately 0.838 euro. E.4156 Consider the figure below o where the dimensions are given in cm and the areas in cm 2 . ABCD is a rectangle. The triangle DCF is a rectangle at D . Part A 1 In this question, we have : AB = 4 ; AF = 6 ; DF = 2 a Calculate the area of the rectangle ABCD . b Calculate the area of triangle DCF . 2 In the following : AB = 4 ; AF = 6 ; DF = x ; AD = 6 − x a Show that the area of rectangle ABCD is 24 − 4 x . b Show that the area of triangle DCF is 2 x . c Solve the equation : 24 − 4 x =2 x . For what value of x is the area of the rectangle ABCD equal to the area of the triangle DCF ? Part B 1 We note f the function defined by: f ( x )=24 − 4 x and g the function defined by: g ( x )=2 x Complete the table below, then graph the function f on the document below on which the graphical representa-tion G of the function g is shown. x 0 1 2 f ( x )=24 − 4 x 2 By graphical reading, determine for what value of x the area of DCF is equal to 6 cm 2 . 3 By graphical reading, determine the area of ABCD for x =2.5 cm . 4 By graphical reading, find the result from question 2 c in part A . For questions 2 , 3 and 4 we will leave the necessary lines visible on the graph. https://chingmath.fr sacados/2483 sacados/4156 ABCDF4x6 x012345678y2468101214161820222426
AgdeSèteMontpellierNîmesBalarucD31D31A927kmBalarucA990kmNîmesD3154kmNîmesD31A9Balaruc14km ABCDSMpignon nord de l’atelier ABSplanches du bardagetasseauxde boisplanches du bardagetasseauxde boisplanches du bardagetasseauxde boisplanches du bardagetasseauxde boisplanches du bardagetasseauxde bois 6. Complex tasks E.6337 Mariam would like to move from Agde to Nimes. On the internet, she manages to retrieve the following infor-mation : Information 1 Information 2 Here is a summary table of speed limits in France : Conditions normales de circulatio Par temps de pluie ou autres précipitations Visibilité inférieure à 50 m Autoroute 130 km = h 110 km = h 50 km = h Voie rapide 110 km = h 100 km = h 50 km = h Autres routes 90 km = h 80 km = h 50 km = h Agglomération 50 km = h 50 km = h 50 km = h Information 3 Here are some features on the gas mileage of Mariam’s car Vitesse (en km = h ) 30 40 50 60 70 80 90 100 110 120 130 Consommation (en ‘ ) 7 7.5 8 6 6.5 7 7.5 7.8 8 8.5 9.2 For safety reasons, Mariam always drives 10 km = h below the speed limit and for budgetary reasons, Mariam always chooses the most economical route. Which route will Mariam choose? E.4339 Monsieur Duchêne wants to barder (cover) the north gable of his workshop with wood. This gable has no openings. We give : AD =6 m ; AB =2.20 m SM =1.80 m M is the middle of [ BC ] . Parts I , II and III are indépendantes Part I 1 Show that the area of the gable ABSDCD of the work-shop is 18.6 m 2 . 2 The wooden planks that will be used to clad the gable are packaged in batches. One lot covers an area of 1.2 m 2 . a How many lots must Mr Duchêne buy at least? b To make sure he doesn’t run out of wood, Mr. Duchêne decides to buy 18 lots. A lot is sold for 49 e . How much should Mr. Duchêne pay? c Monsieur Duchêne has received a discount of 12 % on the amount to be paid. In the end, how much did Mr. Duchêne pay? Part II First, Mr. Duchêne will need to attach wooden cleats to the wall. Next, he will place the siding boards on the cleats, as shown in the figure opposite. The cleats will be placed parallel to the [ AB ] . side The purpose of this part is to determine the length of each cleat according to the distance sepa-rating it from the [ AB ] side Let E be a point on segment [ AD ] . The parallel to ( AB ) passing through E intersects [ BS ] at F and [ BM ] at H . It is assumed that the line ( FH ) is parallel to the line ( SM ) . Segment [ EF ] represents a cleat to be fixed. https://chingmath.fr sacados/6337 AgdeSèteMontpellierNîmesBalarucD31D31A927kmBalarucA990kmNîmesD3154kmNîmesD31A9Balaruc14km sacados/4339 Pondichery Juin 2011 ABCDSMpignon nord de l’atelier ABSplanches du bardagetasseauxde boisplanches du bardagetasseauxde boisplanches du bardagetasseauxde boisplanches du bardagetasseauxde boisplanches du bardagetasseauxde bois
ABCDSM1;80m2;20m0;50m6mFHE ABCDSM1;80m2;20mx6mFHE ABCDSMpignon nord de l’atelier x00,511,522,533,5y0,511,522,533,544,5 1 Knowing that M is the middle of [ BC ] , calculate BM . 2 In this question, it is assumed that the cleat [ EF ] is placed 0.50 m from the [ AB ] side. So we have : AE = BH =0.50 m . a Placing yourself in the triangle SBM and using Thales’ theorem, calculate FH . b Deduce the length EF of the cleat. 3 In this question, we generalize the problem and assume that the cleat [ EF ] is placed at a distance x from the [ AB ] side. So we have : AE = BH = x (with x varying between 0 and 3 m ) a Show that : FH =0.6 x . b Deduce the expression of EF as a function of x . 4 In this question, use the graph in the appendix, which gives the length of a cleat as a function of the distance x separating it from the side [ AB ] . Traces used for graphic readings should be left visible. a What is the length of a cleat knowing that it was placed at 1.50 m from the [ AB ] side? b We have a 2.80 m long cleat that we don’t want to cut. How far from the [ AB ] side should it be placed? Part III Mr. Duchêne needs to know the measure of the angle ∠ SBM to make cer-tain cuts. Recall that : SM =1.80 m ; BC =6 m Determine the measure of the angle ∠ SBM . Round the result to the nearest de-gree. https://chingmath.fr ABCDSM1;80m2;20m0;50m6mFHE ABCDSM1;80m2;20mx6mFHE ABCDSMpignon nord de l’atelier x00,511,522,533,5y0,511,522,533,544,5
ABCDEtagère no1Etagèreno2 ABCDEtagère no1Etagèreno2EFGH E.4340 Part 1: Installing a computer in a school library At the school library, there are two shelves placed in a corner of the room, as shown in the diagram below. To install a computer we move the two shelves the same dis-tance to place a table with the shape AEFGH as in the diagram to the right : We specify that : BE = CF = CG = DH ; GCF is a right triangle and isosceles at C . 1 If we move the two shelves 1 meter, then how long is GF ? 2 In this question, any trace of research, no matter how incomplete, will be considered in the evaluation. We want to have GF =1 m . How much should the shelves be moved? Part 2: Purchase of library management softwareque The school decides to test a software program to manage its library. It downloads this software from the Internet. 1 The file size is 3.5 Mo (megabytes) and the download takes 7 seconds. What is the speed of the internet connection? We will give the result in Mb=s . After a trial period of 1 months, the school decides to pur-chase the software. There are three price points : Rate A : 19 e Rate B : 10 cents per student Rate C : 8 e + 5 cents per student 2 Copy and complete the following table : Nombre d’élèves 100 200 300 Tarif A 19.00 e Tarif B 30.00 e Tarif C 18.00 e 3 a If x represents the number of students, which of the following functions corresponds to the C rate? x ↦−→ 8+5 x ; x ↦−→ 8+0.05 x ; x ↦−→ 0.05+8 x b What is the nature of this function? 4 On the graph given in the appendix, the rate B was plot-ted. On this same graph, plot the rates A and C . The necessary plots for the graphical reading will be shown on the accompanying sheet. 5 By graphical reading, at what number of students is the A rate more attractive than the C rate? In the school, there are 209 students. 6 What is the most attractive rate for the school? Part 3: How the library works Using the software, we can obtain accurate information about the borrowing by 209 students in the school. For example, we have the following data : Nombre d’emprunts en novembre 2010: 0 1 2 3 4 5 6 7 8 Nombre d’élèves : 39 30 36 23 20 22 18 10 11 1 What is the average number of loans per student? 2 What is the median of this series? Part 4: Holiday Party At the end of the school year, the school decides to give read-ing packages to students. 1 Etienne received a package. This package contains 3 comic books and 2 albums. He randomly pulls a first book out of the package without looking. What is the probability that it is a comic book? 2 Etienne took out an album on the first draw. Since he wants to read a comic book, he randomly takes out a second book from the package without looking. What is the probability that it is a comic book? https://chingmath.fr chapExoCorrec/4340 sacados/4340 ABCDEtagère no1Etagèreno2 ABCDEtagère no1Etagèreno2EFGH
x020406080100120140160180200220240260280300320y24681012141618202224262830323436384042TarifB E.5186 The three parts of this problem are independent of each other In a middle school in Caen (Normandy) an exchange program with Mexico is organized for students 3 e who are studying Spanish as a second language. Part A - Student registration The table below shows the distribution of second languages studied by 320 students in 4 e and 3 e at this middle school. Second language studied 4 e 3 e Total Spanish 84 German 24 Italian 62 50 Total 168 320 1 How many students may fi be involved in this exchange? 2 24 students will participate in this trip. Is it true that this represents more than one-fifth of the students in 3 e ? Part B - Financing In order to finance this exchange, two activities are being organized : a Mexican meal and a raffle. 1 The Mexican meal, whereèach participant pays 15 e . On the menu is a typical Mexican dish, chili con carne . Recipe for 4 people 50 g butter 500 g ground beef 2 large onions 65 g tomato paste 2 garlic cloves 400 g red beans 30 cl beef broth 50 people are participating in this meal: a Give the quantity of ground beef, red beans, onions, and tomato paste. b The cost of this meal is 261 e , What is the profit? 2 For the raffle, 720 tickets were sold at a price of 2 e each. The total cost of the prizes to the organizers was 120 . Show that the profit made from these two activities amounts to 1809 . e . Part C - the field trip During their trip to Mexico, the 24 students visited the Mayan pyramid of Chichén Itzá. Access to the temple at the top of the pyramid is via a very steep staircase with 90 steps. The steps are all identical, mea-suring 27 cm wide and 30 cm high. The pyramid of Chichén Itz is shown schematically belowá: https://chingmath.fr x020406080100120140160180200220240260280300320y24681012141618202224262830323436384042TarifB chapExoCorrec/5186 sacados/5186
ABCDEFGHI ABCDEFGH1m5m3m4m5mLe schéma ci-contre n’est pas réalisé à l’échelle ABCD 1 a Justify the following measurements : AB = 24.3 m ; BC = 27 m b Determine the angle formed by the staircase in relation to the ground, rounded to the nearest degree 2 The top of the temple (symbolized by point E ) was built in line with the staircase ramp. Knowing that you have to walk 8.1 m to get from the top of the stairs (point C ) to the entrance of the temple (point G ) , give the total height of this pyramid, rounded to the nearest meter. E.7637 Mr. Chapuis wants to change the tiles and baseboards in the living room of his apartment. To do this, he needs to buy tiles, glue and wooden skirting boards that will be nailed down. He has the following documents : Document 1 : plan the room corresponds to the shaded area. Document 2 Carrelage Size of a tile: 50 cm × 50 cm Thickness of a tile: 0.9 cm Packaging: 1.25 m 2 per box. Price : 19.95 e per box Plinthe Shape : rectangular in length 1 m . Sold by the unit. Price : 2.95 e the wood baseboard. Document 3 Tile adhesive Packaging: bag of 25 kg Yield (area that can be glued) : 4 m 2 per bag Price : 22 e the bag Pack of nails for plinthes Price : 5.50 e per pack 1 a Noting that the length GD is equal to 7 m , deter-mine the area of the triangle BCH . b Show that the area of the room is 32 m 2 . 2 To avoid running out of tile and glue, the vendor advises Mr. Chapuis to allow for an area 10 % larger than the area calculated in question 1 . Mr. Chapuis needs to buy whole boxes and whole bags. Determine the number of boxes of tile and the number of bags of glue to be purchased. 3 The seller also recommends taking a margin of 10 % on the length of the baseboards. Determine the total num-ber of baseboards Mr. Chapuis must purchase to go around the room. We specify that there are no baseboards on the door. 4 How much does Mr. Chapuis have to spend, knowing that he can make do with a package of nails? Round the answer to the nearest euro. 7. Rotation E.868 A pavement is made up of rhom-buses all identical to the rhombus ABCD as the coded figure below. We call R the rotation of center C that transforms D into B . We call t the vector translation 2 −−→ BC . We call S B the symmetry of center B . https://chingmath.fr ABCDEFGHI sacados/7637 ABCDEFGH1m5m3m4m5mLe schéma ci-contre n’est pas réalisé à l’échelle chapExoCorrec/868 sacados/868 Afrique de l'Ouest, Asie - Juin 2005 ABCD
OLMH(d ABCD(d 1 a What is the angle of rotation R ? Justify the answer. b On the figure, draw, in color, the image L 1 of the rhom-bus ABCD by R . 2 On the figure, draw, in color, the image L 2 of the rhom-bus ABCD by t . 3 On the figure, draw, in color, the image L 3 of the rhom-bus ABCD by S B . E.861 1 Draw a circle C of center O and radius 6 cm. Using a straightedge and compass, draw a regular hexagon ABCDEF inscribed in this circle. 2 Note I the midpoint of segment [ AB ] . Calculate the length AI . Justify. (round to the nearest millimetre) Give the length of the sides of this polygon. 3 What can you say about the image of ABCDEF by a rotation of center O and angle 60 o , 180 o , 780 o clockwise or counterclockwise? For what angles is the polygon ABCDEF invariant? 4 Explain why the triangle ACE is an equilateral triangle. E.862 On the grid below, a casserole dish is shown in black. 1 Construct C 1 , the symmetrical of C with respect to the line ( d ) . 2 Construct C 2 , the symmetric of C with respect to the point L . 3 Construct C 3 , the translate of C by the translation that sends the point H to M . 4 Construct C 4 , the image of C by the rotation with center O , angle 90 o and counterclockwise direction. E.863 We always start with the figure shown below : 1 Draw in blue its symmetrical with respect to the point C 2 Rotate 90 o of the figure in green with respect to the point D clockwise. 3 Draw its symmetric with respect to the line ( d ) in red. 4 By the translation that transforms the point A into the point B , perform the translation of the figure in black. https://chingmath.fr chapExoCorrec/861 sacados/861 chapExoCorrec/862 sacados/862 OLMH(d chapExoCorrec/863 sacados/863 ABCD(d
ABCDEIO5;7cm54o ATYUHLGNO45o ABIE E.2648 Here is the regular pentagon ABCDE . Point I is the midpoint of [ AB ] . OA = OB = OC = OD = OE = 5.7 cm 1 a What is the nature of triangle AOB ? b Show that the measure of angle angle AOB is 72 o . 2 What is the image of triangle BOC , a by the axial symmetry of axis ( DI ) ? b by rotation around the center O , with an angle of 72 o , counterclockwise? 3 Calculate the length AB , rounded to the nearest millime-ter. E.3953 Below is shown HUY TANGL a regular octagon with center O . 1 What is the symmetric of T by central symmetry of cen-ter O ? 2 What is the symmetric of T about the ( GY ) axis? 3 What is the image of T by the rotation of center O and angle 135 o clockwise. E.4022 In the figure opposite : BE =4 cm ; I is the middle of segment [ BE ] . A is a point on the circle of diameter [ BE ] such that the measure of the an-gle ∠ BEA is 60 o 1 Reproduce the figure full size on the copy. 2 A is the image of B by a rotation with center I . Specify the measure of the angle of this rotation, justifying your answer. 3 We call F the symmetrical of A with respect to the point I . Determine the nature of the triangle BIF . 8. Unclassified exercises E.2355 RLK is a right-angled triangle in R , with RK =6 cm and RL =9 cm . M is any point on side [ RK ] . We pose RM = x cm . P is the point on segment [ RL ] such that : RP = RM = x . We then place the point N so that RMNP is a square. 1 In this question x =2 . We obtain the following figure (note that the point N lies inside the triangle RKL ) . https://chingmath.fr chapExoCorrec/2648 sacados/2648 Brevet - Polynesie - Juin 2008 ABCDEIO5;7cm54o chapExoCorrec/3953 sacados/3953 ATYUHLGNO45o chapExoCorrec/4022 sacados/4022 ABIE sacados/2355
RKLMPNA1B1C1 a Calculate the area of the triangle RKL . b Calculate the area A 1 of the square RMNP . Calculate the area B 1 of the triangle KMN . Calculate the area C 1 of triangle NPL . c Calculer A 1 + B 1 + C 1 . Check that the area of the quadrilateral RKNL is less than the area of the tri-angle RKL . 2 In this question x =5 . a Make an accurate figure. b Where is the point N now in relation to the triangle RKL ? c We now call A 2 the area of the square RMNP , B 2 the area of triangle KMN and C 2 the area of triangle NPL . Calculate these three areas and check that the area of RKNL is greater than that of triangle RKL . 3 We now take any x a Calculate the area A 3 of the square RMNP as a func-tion of x . Calculate the area B 3 of the triangle KMN as a func-tion of x . Calculate the area C 3 of triangle NPL as a function of x . b Show that : A 3 + B 3 + C 3 = 15 x 2 c We investigate whether there is a value x for which the point N lies on the segment [ KL ] . To do this, solve the equation obtained by writing: A 3 + B 3 + C 3 = Aire du triangle RKL Conclude. E.2682 For each row in the table below, 3 answers are provided, but only one is correct. Find the correct answer and write the corresponding number in the right column. Details of the calculations are not required on the copy. Réponse a Réponse b Réponse c 1 3 2 + 11 5 × 15 2 est égal à 111 4 18 35 2 2 14 × 10 7 × 27 × 10 3 21 × 10 2 1800 18000000 18000 3 The number (30 2) 2 is equal to : 60 3600 1800 4 For any number x , (5 x − 2) 2 is equal to : 5 x 2 - 20 x +4 25 x 2 - 4 25 x 2 - 20 x +4 5 The équation (2 x − 3)( x +4)=0 admits as solutions : 2 3 and − 4 3 2 and − 4 − 3 2 and 4 6 An object costs 12000 F . Its price increases by 5 % . What will its new price be? 12600 F 12500 F 11400 F 7 A car is traveling at a speed of 50 km=h . In how long does it travel 110 kilometers? 2 h 20 min 2 h 12 min 60 min https://chingmath.fr RKLMPNA1B1C1 sacados/2682
ABCH ABCDE40m20m50m 6cm E.3894 The results given in some of the questions can be used to contienue the problem. Throughout the exercise, the unit of length is centimeters. ABC is a triangle such that : AB = 6 cm ; BC = 10 cm ; ∠ ABC = 120 o The height from A intersects the line ( BC ) at H . The figure below is not full-scale 1 Draw the figure at full size. 2 a Calculate the measure of the angle ∠ ABH . Deduce that : BH = 3 cm . b Prove that AH =3 3 , then calculate the area of the triangle ACH (The exact value will be given) c Prove that : AC =14 . 3 M is a point on segment [ BC ] such that : CM =6.5 . The parallel to ( AH ) passing through M intersects the segment [ AC ] at N . a Complete the figure. b Prove that : NM = 3 3 2 . c So that this question, any trace of research, even in-complete, will be considered in the evaluation. Determine the area of the trapezoid AHMN . Give a value approximated to the nearest unit for this area. E.5185 Peter has just bought a piece of land whose shape can be compared to the figure below : He wants to put grass on the entire lot. To do this, he wants to buy a product that comes in a bag of 15 kg où it is written ˇ 1 kg for 35 m 2 ı. 1 How many bags of grass will he need to buy? 2 In addition, he would like to screen the contents of his yard. He has 150 m of wire mesh is this enough? Justify. 3 a In the figure in appendix , draw the circumscribed circle of the triangle BCD with the compass and un-sharpened ruler. b Peter decides in the half-disk outside his lot to plant orange trees. Determine the measurement, exact and approximated to the nearest meter-square, of the area of this planting. E.6396 This figure consists of a square inscribed with four circles tangent to each other and to the sides of the square. Determine the measure of each of the radii in this fig-ure. https://chingmath.fr sacados/3894 Portugal Juin 2010 ABCH sacados/5185 ABCDE40m20m50m chapExoCorrec/6396 sacados/6396 Sangaku 6cm
CollègeMarchéGendarmerieEcoleHorlogeMairiePompierStadePlaceCathédraleLycéeConservatoirePiscineBibliothèque ABC32o2;3kmEFG25o5;4m ABCDEFGHIJKLMNOPQRSTUVWX ABC7cm24cm25cm x012345y123Cf E.7632 Here is a map of two bus lines : It is at 6 h 30 that the two 1 and 2 line buses depart from the ˇ Mairie ı stop in a clockwise direction. The 1 line bus takes 3 minutes between each stop (including parking time) , while the 2 line bus takes 4 minutes. Both will make the full circuit a large number of times. They will stop just after 20 h . Will both buses meet at some point during the day at the ˇ City Hall ı stop at the same time? If so, give all the specific times of these meetings. E.10977 Consider the two triangles shown be-low : 1 Determine the length of segment [ AC ] rounded to the nearest metre. 2 Determine the length of segment [ EF ] rounded to the nearest metre. E.10978 This exercise is a multiple-choice ques-tionnaire. (QCM) . For each question, three answers (A, B, or C) are provided. Only one answer is correct. Copy the question number and the letter corresponding to the correct answer onto your answer sheet. No justification is required. a b c 1 What is the image of point O under the translation that transforms X into J ? A I Q 2 In the right-angled triangle ABC at B , determine the value of cos ACB . 0.28 0.29 0.96 3 The graph C f representing the func-tion f is given in the coordinate system below. Give the value of the antecedent of the number 1 . 1 2 4 4 The expression x − 3 2 − x 2 − 9= ::: 0 − 6 x − 9 x 5 Consider the function f defined by: f ( x ) = x 2 − 5 x + 2 What is the image of − 4 under the func-tion? -34 6 38 6 The heights, in meters, of six students were measured : 1.64 ; 1.53 ; 1.49 ; 1.56 ; 1.62 ; 1.64 What is the average height, in meters, of these students? 1.58 1.59 1.6 7 The number of divisors of the integer 26 is 4 6 8 https://chingmath.fr sacados/7632 CollègeMarchéGendarmerieEcoleHorlogeMairiePompierStadePlaceCathédraleLycéeConservatoirePiscineBibliothèque chapExoCorrec/10977 sacados/10977 ABC32o2;3kmEFG25o5;4m chapExoCorrec/10978 sacados/10978 ABCDEFGHIJKLMNOPQRSTUVWX ABC7cm24cm25cm x012345y123Cf