Grade 8 / Pythagoras' theorem 57 exercises corrected

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ChingQuizz : 11 exercises available for Quizz assessment : 1. Problem-solving situations E.5798 The figure below consists of a ABC triangle with a right angle of C . On each side of the triangle ABC is constructed a square : Cut out the five parts shown, in the figure, in dotted lines and find the ˇ good ı paving with these five parts covering the entirety of the large hatched square. An animation showing the process to be performed E.5799 As shown in the previous exercise, cutting one of the squares and overlapping the largest square with the other two squares is feasible for all right-angled triangles as shown in the ap-plication opposite. We’ll admit this fact for this exercise. Consider the triangle ABC right-angled C represented below with : CB = 3 m ; CA = 4 m Has been represented in dotted line, the cutting of the square of side [ AC ] allowing the overlap of the square of side [ AB ] by the other two squares : 1 Determine the area A 1 of the square with side [ AC ] . 2 Determine the area A 2 of the square with side [ BC ] . 3 a Deduce the area A 3 of the square with side [ AB ] . b Deduce the measure of side [ AB ] . https://chingmath.fr quizzChap/1036 chapExoCorrec/5798 sacados/5798 ABC r225-11 chapExoCorrec/5799 sacados/5799 r225-11 ABC4m3mA1A2A3
E.8699 Below are constructed, externally to the ABC triangle, the squares whose sides are those of the ABC triangle: It is assumed that the proposed cutting of the two shaded squares with sides [ AC ] and [ BC ] provides a perfect overlap of the shaded square with side [ AB ] . Determine the exact measure of the segment [ AB ] . Illustration : The following video demonstrates the use of such a cutout. E.5800 The triangle ABC below is right-angled at C . It is assumed that the proposed cutout of the shaded squares perfectly and completely covers the hatched square. Determine the length of side [ BC ] . 2. Introduction to square roots E.8693 Definition: for any number a , the number obtained by multiplying the number a by itself is called square of the number a . This number is denoted a 2 . Examples: 3 2 = 9 ; 7 2 = 49 Complete the statements below : a The square of the number 4 is . . . . . . b The number . . . . . . has square 36 c The square of the number 7 is . . . . . . d The number . . . . . . has the square 16 E.1766 For each question, add the number et = ou to complete the missing operation. 1 2 The operation marked 2 represents the square of . Definition: let a be a positive or zero number. We call the square root of a the single positive number whose square is a . We note this number a . Examples: (related to previous exercise) : The number whose square is 9 is 3 : 64 = 8 4 = 2 25 = 5 144 = 12 1.44 = 1.2 https://chingmath.fr chapExoCorrec/8699 sacados/8699 ABC3;2cm2;4cm r225-11 chapExoCorrec/5800 sacados/5800 ABC9;1cm10;9cm chapExoCorrec/8693 sacados/8693 chapExoCorrec/1766 sacados/1766 ×2?5??−2??5?÷2 ?3?×4−5???−2;2?;4 23?28?2?4 2?252?1442?1;44
E.8691 Say whether the statements below are true or false : 1 The square of the number 5 is 10 . 2 The square root of the number 3 is 9 . 3 The number 25 has square root 5 . 4 The square root of the number 1 000 has value 100 . E.8956 Complete the statements below : 1 The square root of the number 81 is . . . . . . 2 The number 4 is the square root of the number . . . . . . 3 The square root of the number 0 has the value . . . . . . 4 The square of the number 3 has the square root . . . . . . E.8731 Give the value of the sum : of the square root of 25 and the square of ( − 2) Indication : we will also indicate the value of these two terms E.8732 Give the value of the sum : of the square root of 9 and the square of ( − 4) Indication : we will also indicate the value of these two terms 3. Pythagorean triples E.1768 Definition: a triplet of numbers ( a ; b ; c ) is said to be a Pythagorean triplet if they verify the relation: a 2 + b 2 = c 2 Example: the triplet 3 ; 4 ; 5 is a Pythagorean triplet: 3 2 + 4 2 = 9 + 16 = 25 ; 5 2 = 25 The triplet 4 ; 5 ; 6 is not a pythagoric triplet because : 4 2 + 5 2 = 16 + 25 = 41 ; 6 2 = 36 Note: these triplets of numbers are important in geometry as we will see in this chapter. For each row of the table below, we wish to determine the value of the positive number c verifying the following rela-tionship : a 2 + b 2 = c 2 a b a 2 b 2 a 2 + b 2 c 2 c 4 3 8 6 15 8 5 12 Hint: to complete the last column, we can use the square root key your calculator or the list of the first twenty perfect squares : 0 0 = 4 1 6 ; 8 6 4 = 12 1 44 ; 16 2 56 = 1 1 = 5 2 5 ; 9 8 1 = 13 1 69 ; 17 2 89 = 2 4 = 6 3 6 ; 10 1 00 = 14 1 96 ; 18 3 24 = 3 9 = 7 4 9 ; 11 1 21 = 15 2 25 ; 19 3 61 = E.8692 For each row in the table below, we wish to determine the value of the positive number c verifying the following relationship : a 2 + b 2 = c 2 a b c a 2 b 2 a 2 + b 2 = c 2 4 5 8 17 12 13 30 50 E.1767 Check whether the numbers a , b , c form a Pythagorean triplet. If so, write the equality found in the last column : a b c a 2 b 2 c 2 Egalité trouvée 3 5 4 27 36 45 6.5 3.3 5.6 16 10 20 3.5 9.1 8.4 10 2 10.1 https://chingmath.fr chapExoCorrec/8691 sacados/8691 chapExoCorrec/8956 sacados/8956 chapExoCorrec/8731 sacados/8731 chapExoCorrec/8732 sacados/8732 chapExoCorrec/1768 sacados/1768 chapExoCorrec/8692 sacados/8692 chapExoCorrec/1767 sacados/1767
E.1784 Complete the table below to obtain the value of the number c verifying the requested equality for each row : a b a 2 b 2 Egalité c 2 c 12 5 a 2 + b 2 = c 2 7 25 a 2 + c 2 = b 2 5.8 4.2 b 2 + c 2 = a 2 5.6 11.9 c 2 + a 2 = b 2 4. Introduction to the Pythagorean theorem E.1765 Consider the two triangles ABC and DEF rectangles respectively D and A shown below : 1 Complete the table below ; the first row is to be com-pleted by taking measurements on the triangles : AB BC AC ED DF EF x x 2 2 For each triangle, write an expression of the form : a 2 + b 2 = c 2 where the numbers a , b , c represent the lengths of the triangle. E.1748 Consider the two triangles ABC and DEF shown below : A graduation per half unit is provided on the sides of these triangles. 1 a Taking the measurements of these triangles, com-plete the table below : AB BC AC ED DF EF x x 2 b For each of the triangles, check whether the lengths of their sides define a Pythagorean triplet. 2 a Check, using the square, whether the angles ACB and DFE are right angles. b Make a conjecture between the nature of the triangle and the nature of the triplet of triangle lengths. Definition: a conjecture is a proposition made, or the consequence of an observation, which is assumed to be true but for which we have no proof . https://chingmath.fr chapExoCorrec/1784 sacados/1784 chapExoCorrec/1765 sacados/1765 ABCDEF chapExoCorrec/1748 sacados/1748 ABCDEF1cm
E.4476 Consider the triangle ABC right-angled C shown below : whose sides [ AB ] , [ BC ] , [ AC ] have measures b , a , c , respec-tively. We denote ¸ and ˛ as the respective measures of the angles ∠ CAB and ∠ CBA . From the triangle ABC , we construct a square of side a + b shown in Figure 1 . y Then use 4 triangles identical to the ABC triangle to produce, in each case, figures 2 and 3 These two figures help define the three quadrilaterals Q 1 , Q 2 , and Q 3 shown above. 1 Justify that the quadrilaterals Q 1 , Q 2 , Q 3 are squares with sides measuring c , a , and b respectively. 2 Below is shown the triangle ABC on the outside of which has been constructed a square from each of its sides. In-side each of these squares, is noted their areas. Justify the equality: a 2 + b 2 = c 2 The Pythagorean theorem can be proved in several ways. Here is one of them : E.11574 In the three cases below, consider a tri-angle ABC whose three lengths are known : 1 Using Geogebra, construct each of these triangles and, by measuring angle ACB , check whether the triangle is a right triangle. 2 Complete the table below : Is there a relationship between the squares of the side lengths in the case where the triangle is right-angled? https://chingmath.fr chapExoCorrec/4476 sacados/4476 ABCbca¸˛ abcabFigure1 Q1Figure2 Q2Q3Figure3 c2a2b2 r536-0 r536-0 sacados/11574 Cas 1Cas 2Cas 3ABCABCABC6;4cm2;5cm6cm6;5cm2;5cm6cm6;6cm2;5cm6cm Cas1Cas2Cas3ABACBCABACBCABACBCRectangle?LongueurCarrédelalongueur
E.11575 In the three cases below, consider a tri-angle ABC whose three lengths are known : 1 Using Geogebra, construct each of these triangles and, by measuring angle ACB , check whether the triangle is a right triangle. 2 Complete the table below : Is there a relationship between the squares of the side lengths in the case where the triangle is right-angled? 5. Pythagorean theorem and deductive steps E.8696 Pythagorean theorem: If a triangle is right-angled then the square of the length of the mortgage is equal to the sum of the squares of the lengths of the other two sides. Definition: Let ABC be a right-angled triangle in A . We call equality of the Pythagorean theorem in the tri-angle ABC the relation: BC 2 = AB 2 + AC 2 Note: in Pythogoras’ equality, previous the side [ BC ] is the hypotenuse of the right triangle ABC . It is also the longest side of this triangle. Consider the triangle ABC right-angled at C and verifying: CA = 6 m ; CB = 1.1 m Using the deductive chains below determine the measurement of side [ AB ] : E.8700 Consider the triangle DEF right-angled D and verifying: EF = 5 m ; DF = 4.8 m Using the deductive chains below determine the measurement of side [ DE ] : https://chingmath.fr sacados/11575 Cas 1Cas 2Cas 3ABCABCABC6;4cm2;5cm6cm6;5cm2;5cm6cm6;6cm2;5cm6cm Cas1Cas2Cas3ABACBCABACBCABACBCRectangle?LongueurCarrédelalongueur chapExoCorrec/8696 sacados/8696 JesaisJ’utiliseJ’endéduisD’après le théorème de Pythagore222Chaînonsdéductifs 222AB2ABABCalculs chapExoCorrec/8700 sacados/8700 JesaisJ’utiliseJ’endéduisD’après le théorème de Pythagore222Chaînonsdéductifs 2222−DE2DEDEDE2Calculs
6. Pythagorean theorem E.1061 For each triangle, determine, if pos-sible, the unknown length. E.1062 In each of the triangles below, de-termine the unknown length. E.1806 For each triangle and if possible, determine the unknown length. E.8777 Consider the right triangle ABC with right angle at C such that : AB =26 m ; BC =24 m Determine the length of seg-ment [ AC ] . Reminders: 1 2 =1 ; 9 2 =81 ; 17 2 =289 ; 25 2 =625 2 2 =4 ; 10 2 =100 ; 18 2 =324 ; 26 2 =676 3 2 =9 ; 11 2 =121 ; 19 2 =361 ; 27 2 =729 4 2 =16 ; 12 2 =144 ; 20 2 =400 ; 28 2 =784 5 2 =25 ; 13 2 =169 ; 21 2 =441 ; 29 2 =841 6 2 =36 ; 14 2 =196 ; 22 2 =484 ; 30 2 =900 7 2 =49 ; 15 2 =225 ; 23 2 =529 ; 31 2 =961 8 2 =64 ; 16 2 =256 ; 24 2 =576 ; 32 2 =1024 E.1072 Consider the triangle DEF right-angled E such that : DE =6 cm ; EF = 6.3 cm Determine the measurement of side [ DF ] . E.8733 Consider the triangle ABC right-angled C such that : AB =13 cm ; AC =6.6 cm Determine the measure of seg-ment [ BC ] . E.6181 Consider the triangle ABC right-angled A shown below such that : AB =7.2 cm ; BC = 9 cm Determine the length of side [ AC ] . E.6182 Consider the triangle ABC right-angled A shown below such that : AB =8.4 cm ; BC = 11.6 cm Determine the length of side [ AC ] . E.8734 Following a tornado, a wooden pole broke. Below is a picture of the broken pole: Determine the height of the pole before the tornado. https://chingmath.fr chapExoCorrec/1061 sacados/1061 ABC40km30kmDEF12cm13cmGHI5m3m chapExoCorrec/1062 sacados/1062 DEF75m45m32m24mBAC chapExoCorrec/1806 sacados/1806 7;5cm7;2cmEDF32m24mBAC chapExoCorrec/8777 sacados/8777 ABC24m26m chapExoCorrec/1072 sacados/1072 EDF6;3cm6cm chapExoCorrec/8733 sacados/8733 ABC13cm6;6cm chapExoCorrec/6181 sacados/6181 ABC7;2cm9cm chapExoCorrec/6182 sacados/6182 ABC8;4cm11;6cm chapExoCorrec/8734 sacados/8734 15m8m
E.8735 Following a spin, a wooden post broke. Below is a picture of the broken pole: Determine the height of the pole before the spinning. 7. Using the Pythagorean theorem twice E.4439 In the figure below, ABCD is a rectangle and ACE is a right triangle at A . 1 a Justify that the triangle ADC is a right triangle at D . b Determine the measure of segment [ AC ] . 2 Determine the measure of segment [ EC ] . E.4440 In the figure below, ABCD is a rectangle and ACE is a right triangle at E . 1 a Justify that the triangle ACD is right-angled D . b Determine the length of segment [ AC ] . 2 Determine the length of segment [ EC ] . 8. Approximate measurements E.1060 Figures are not drawn to actual di-mensions. For each of the triangles, determine the length of the segment [ AB ] , to the nearest tenth of a centimeter: a b E.4421 Consider the polygon ABECD rep-resenting a farmer’s field : Determine the length of the fence in this field, rounded to the nearest meter, when x =30 m . https://chingmath.fr chapExoCorrec/8735 sacados/8735 16m20m chapExoCorrec/4439 sacados/4439 2;8cm2;1cm1;2cmABCDE chapExoCorrec/4440 sacados/4440 4cm3cm4;8cmABCDE chapExoCorrec/1060 sacados/1060 15cm4cmABC 1cm1cmABC chapExoCorrec/4421 sacados/4421 ABCDEx
E.8698 Consider the figure below where the triangles ABC , CDA , EFG and GFH are right-angled tri-angles : Establish that the segments [ AD ] and [ FH ] are the same length. E.1858 Without any justification, give the exact measure of the lengths of the segments [ OB ] , [ OC ] and [ OD ] . 9. Areas E.4478 The figure below shows a right tri-angle ABC in A and a quadrilateral BCDE . 1 Determine the length of segment [ BC ] to the nearest mil-limeter. 2 a What is the nature of the quadrilateral BCDE ? Jus-tify your answer. b Determine the area of the quadrilateral BCDE . c Determine the perimeter of polygon ABEDC to the nearest millimeter. E.6413 Consider the triangle ABC right-angled C and the point H foot of the height from the vertex C . The following information is available: the segment [ AH ] measures 9 cm ; we have the areas of the following two triangles : A ACH = 54 cm 2 ; A ABC = 150 cm 2 1 Determine the measure of segment [ CH ] . 2 Determine the measure of segment [ BC ] . https://chingmath.fr chapExoCorrec/8698 sacados/8698 16m12m21mBACDEFGH21m12m16m chapExoCorrec/1858 sacados/1858 AO1cmBCD chapExoCorrec/4478 sacados/4478 ABCDE5cm9cm chapExoCorrec/6413 sacados/6413 ABCH9cm
E.6210 Consider the rectangle ABCD such that : AB = 4 cm ; AD = 3 cm Note H the foot of the height from C in triangle DCB . Note I the foot of the height from H in triangle DCH . 1 a Determine the measure of segment [ DB ] . b Determine the measure of the area of triangle DCB . c Deduce that segment [ CH ] has measure 2.4 cm . 2 Determine the measure of segment [ HB ] . 3 a Deduce the measure of segment [ DH ] . b Determine the area of triangle DCH . c Deduce the measure of segment [ IH ] . 1 2 =1 1.1 2 =1.21 1.2 2 =1.44 1.3 2 =1.69 1.4 2 =1.96 1.5 2 =2.25 1.6 2 =2.56 1.7 2 =2.89 1.8 2 =3.24 1.9 2 =3.61 2 2 =4 2.1 2 =4.41 2.2 2 =4.84 2.3 2 =5.29 2.4 2 =5.76 2.5 2 =6.25 2.6 2 =6.76 2.7 2 =7.29 2.8 2 =7.84 2.9 2 =8.41 3 2 =9 3.1 2 =9.61 3.2 2 =10.24 3.3 2 =10.89 3.4 2 =11.56 3.5 2 =12.25 3.6 2 =12.96 3.7 2 =13.69 3.8 2 =14.44 3.9 2 =15.21 E.4477 A forest park is represented below by the rectangle ABCD : To traverse from A to C this forest, a path depicted as dashed has been cut through the park. 1 a Determine the area of triangle ABD . b Determine the length of segment [ BD ] c Relative to the triangle ABD , name the segment [ AM ] . d Deduce the length of segment [ AM ] . It is assumed that the triangles AMD and BCN are of iden-tical dimensions. 2 Determine the total length of the path through this for-est. 10. Solid geometry E.4950 Consider the right-handed paving stone ABCDEFGH shown below whose measurements are known as follows : HG = 10 cm ; HD = 6 cm ; DA = 4.5 cm 1 a What is the nature of the triangle ADH ? b Draw the triangle ADH in real size. c Determine the exact value of the length AH . 2 a What is the nature of the triangle AHG ? b Draw the triangle AHG in real size. c Determine the exact value of the length AG . E.1073 ABCDEFGH is a cube of 3 cm edge. 1 Calculate the length of [ AH ] to the nearest millimetre. 2 a Without justification, give the nature of the quadrilateral ABGH ? b Assume that the triangle BAH is a right-angled tri-angle. Calculate the length of [ AG ] to the nearest mil-limetre. https://chingmath.fr chapExoCorrec/6210 sacados/6210 ABCDH3cm4cmI chapExoCorrec/4477 sacados/4477 ABCDMN32km24km chapExoCorrec/4950 sacados/4950 ABCDEFGH10cm6cm4;5cm chapExoCorrec/1073 sacados/1073 ABCDEFGH
E.4956 Consider the right-handed paving stone ABCDEFGH shown below whose measurements are known as follows : HG = 4.8 cm ; FG = 3.6 cm ; HB = 7.5 cm Determine the exact measurement of the height [ FB ] of this rectangular cuboid. E.4955 Consider the right block ABCDEFGH shown below whose measurements are known as follows : HG = 12 cm ; FG = 8 cm ; FB = 9 cm Determine the length of the segment [ BH ] . 11. Open-ended problems E.6298 In Pisa around 1200 A.D. (prob-lem attributed to Leonardo of Pisa, known as Fibonacci, Italian mathe-matician of the Middle Ages Âge) . A spear, 20 feet long*, is placed ver-tically along a tower considered per-pendicular to the ground. If the end of the spear resting on the ground is moved 12 feet away from the tower, how far does the other end of the spear descend along the wall? * A foot is an Anglo-Saxon unit of measurement worth approximately 30 cm E.5782 A cyclist is at the bottom of the hill and in the middle of one side and wants to reach the middle of the other side. He has two choices : either he goes around the hill or he goes over the top of the hill. This cyclist rides at 32 km = h on a flat road, at 25 km = h on an uphill road, 41 km = h on a downhill road. To facilitate his decision making, we model the hill as a reg-ular square-based pyramid with side 2 km whose height is 225 m . Help him choose the fastest path. E.8697 Opposite, two circles lie inside a square. They are tangent to each other and are each tangent to two sides of the box. Determine the length of the diago-nal of this square. E.8754 On the cylinder shown opposite, an ant located at point G wants to move to point H by following the arrow path shown in the fig-ure. This path is the shortest route around the lateral surface of the cylinder. Determine the length of the path taken by the ant, rounded to the nearest millimeter. Hint: we will use : ı ≈ 3 ; 14 12. Unclassified exercises https://chingmath.fr chapExoCorrec/4956 sacados/4956 ABCDEFGH3;6cm4;8cm7;5cm chapExoCorrec/4955 sacados/4955 ABCDEFGH8cm12cm9cm chapExoCorrec/6298 sacados/6298 TourLance12piedsh chapExoCorrec/5782 sacados/5782 DépartArrivée chapExoCorrec/8697 sacados/8697 1m1m chapExoCorrec/8754 sacados/8754 2cmGH5cm
E.8715 Consider the squareé ABCD shown be-low with side 24 cm Consider the points M , N , P belonging to the sides [ AB ] , [ AD ] , [ BC ] , respectively, and verifying the measures : AM = 16 cm ; MN = 20 cm ; MP = 17 cm Determine the area of the triangle MNP . E.4957 Consider the right-handed paving stone ABCDEFGH shown below whose measurements are known as follows : HG = 6.4 cm ; FG = 4.8 cm ; HB = 10 cm Determine the exact measurement of the height [ FB ] of this rectangular cuboid. E.7628 With a geometry software, we execute the program below. Construction program : Build a square ABCD ; Draw the circle of center A and radius [ AC ] ; Place the point E at the intersection of the circle and the half line [ AB ) ; Construct a square DEFG . Figure obtained: 1 On the copy, perform the construction with AB =3 cm . 2 In this question, AB =10 cm . a Show that : AC = 200 cm b Explain why: AE = 200 cm c Show that the area of the square DEFG is triple the area of the square ABCD . 3 It is assumed for this question that for any length of side [ AB ] , the area of the square DEFG is always triple the area of the square ABCD . Running this construction program, we want to obtain a square DEFG with an area of 48 cm 2 . What length AB should be chosen initially? E.781 The ABCDEFGH cube, shown opposite, has each of its edges measuring 3 cm . 1 Calculate the length FH 2 Calculate length BH 3 Calculate the area of the triangle BFH . Hint: we will round : lengths to the nearest tenth of a millimetre. areas to the nearest millimetre-square. https://chingmath.fr chapExoCorrec/8715 sacados/8715 ABCDMNP17cm20cm24cm16cm chapExoCorrec/4957 sacados/4957 ABCDEFGH4;8cm6;4cm10cm chapExoCorrec/7628 sacados/7628 ABCDEFG chapExoCorrec/781 sacados/781 ABCDEFGH
E.11167 The figure below consists of two squares ABCD and CEFG and a right triangle BCE with right angle at E . Determine the total area of the figure. E.11168 The figure below is composed of two squares ABCD and CEFG and a right triangle BCE with E . Determine the perimeter of this figure. E.6447 Consider the configuration below com-posed of the two triangles ABC and ABD rectangles in C and D respectively: 1 a Determine the measure of the segment [ BC ] . b Give the measure of the angle ABC , rounded to the nearest tenth of a degree. 2 Justify that the line ( BA ) is the bisector of the angle CBD . E.1107 1 Using a compass and a straightedge, perform the follow-ing plotting program : a Draw a triangle ABC equilateral of 3 cm side ; b Trace, in the triangle ABC , the height resulting from B ; c Name H the foot of this height. 2 Determine the value of the angle ABH . Justify. 3 a Give the length of segment [ AH ] . Justify. b Calculate the length of height [ BH ] to the tenth of a centimeter. https://chingmath.fr chapExoCorrec/11167 sacados/11167 ABCDEFG225cm264cm2 chapExoCorrec/11168 sacados/11168 BCEFGHI441cm2841cm2 chapExoCorrec/6447 sacados/6447 ABCD5cm1;4cm chapExoCorrec/1107 sacados/1107