Grade 8 / Reciprocal of the Pythagorean theorem 40 exercises (100% corrected)

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ABC20m21m29mDEF8m15m17m BAC12m9m15m JesaisJ’utiliseJ’endéduisLes longueurs du triangleABCvéri∏ent l’égalitéde Pythagore car:*AB2CA2CB2*BC2AB2AC2*AC2BA2BC2D’après la réciproque du théorème de PythagoreChaînondéductifCalculsAB2:::AC2:::BC2::: ChingQuizz : 10 exercises available for Quizz assessment : 1. Pythagorean equality E.8760 Definition: three numbers form a Pythagorean triplet if the square of one number is equal to the sum of the squares of the other two. Example : the numbers 6 , 8 and 10 form a Pythagorean triplet because : 10 2 = 6 2 + 8 2 Justify, for each question, that these three numbers form a Pythagorean triplet: a 3, 4 and 5 b 5, 12 and 13 c 9, 12 and 15 E.8761 Consider the two triangles ABC and DEF shown in the hand below : 1 Using the calculator, complete the following blanks : a AB 2 = : : : : : : AC 2 = : : : : : : BC 2 = : : : : : : b DE 2 = : : : : : : DF 2 = : : : : : : EF 2 = : : : : : : Definition: a triangle is said to verify the Pythagorean equality if its lengths form a Pythagorean triplet. 2 Justify that the triangles ABC and DEF verify Pythagorean equality. 2. Reciprocal of the Pythagorean theorem and deductive chains E.8759 Reciprocal of the Pythagorean theorem: If, in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides then that triangle is right-angled and that side is its hypotenuse. Another version of the statement: If a triangle verifies the Pythagorean equality then this tri-angle is right-angled and the largest of its sides is its hy-potenuse. Elias wonders if the triangle-shaped wall built along the edge of his garden will form a right angle. His father gave him this sketch of his garden with the exact measurements of the garden : To help Elias, complete this deductive link: https://chingmath.fr chapExoCorrec/8760 sacados/8760 chapExoCorrec/8761 sacados/8761 ABC20m21m29mDEF8m15m17m chapExoCorrec/8759 sacados/8759 BAC12m9m15m JesaisJ’utiliseJ’endéduisLes longueurs du triangleABCvéri∏ent l’égalitéde Pythagore car:*AB2CA2CB2*BC2AB2AC2*AC2BA2BC2D’après la réciproque du théorème de PythagoreChaînondéductifCalculsAB2:::AC2:::BC2:::
BAC0;9cm4cm4;1cm JesaisJ’utiliseJ’endéduisChaînondéductifCalculsAB216;81AC20;81BC216 ABC13cm12cm5cmDEF29m21m20m BAC2km2;1km2;9kmFDE4;2mm5;6mm7mm DEF8;7cm6cm6;3cm E.8762 A ABC triangle has been depicted below freehand but its true measurements are shown : Complete the deductive link below to prove that this triangle is right-angled. We will specify the vertex of the right angle. 3. Reciprocal of the Pythagorean theorem E.1797 Below are freehand drawings of the two triangles ABC and DEF , whose side lengths are indi-cated in the figure : Prove that these triangles are right triangles. Indicate the vertex of the right angle. Reminder : The first perfect squares are: 1 2 = 1 11 2 = 121 21 2 = 441 31 2 = 961 41 2 = 1681 2 2 = 4 12 2 = 144 22 2 = 484 32 2 = 1024 42 2 = 1764 3 2 = 9 13 2 = 169 23 2 = 529 33 2 = 1089 43 2 = 1849 4 2 = 16 14 2 = 196 24 2 = 576 34 2 = 1156 44 2 = 1936 5 2 = 25 15 2 = 225 25 2 = 625 35 2 = 1225 45 2 = 2025 6 2 = 36 16 2 = 256 26 2 = 676 36 2 = 1296 46 2 = 2116 7 2 = 49 17 2 = 289 27 2 = 729 37 2 = 1369 47 2 = 2209 8 2 = 64 18 2 = 324 28 2 = 784 38 2 = 1444 48 2 = 2304 9 2 = 81 19 2 = 361 29 2 = 841 39 2 = 1521 49 2 = 2401 10 2 = 100 20 2 = 400 30 2 = 900 40 2 = 1600 50 2 = 2500 E.1066 Consider the triangles ABC and DEF shown below, drawn freehand, with their exact mea-surements added : Show that these triangles are right triangles. Indicate the vertex of the right angle. E.8783 Consider the triangle DEF shown below and having the measures : DE =8.7 cm ; EF =6.3 cm ; DF =6 cm Show that the triangle DEF is a right triangle with the vertex of the right angle specified. https://chingmath.fr chapExoCorrec/8762 sacados/8762 BAC0;9cm4cm4;1cm JesaisJ’utiliseJ’endéduisChaînondéductifCalculsAB216;81AC20;81BC216 chapExoCorrec/1797 sacados/1797 ABC13cm12cm5cmDEF29m21m20m chapExoCorrec/1066 sacados/1066 BAC2km2;1km2;9kmFDE4;2mm5;6mm7mm chapExoCorrec/8783 sacados/8783 DEF8;7cm6cm6;3cm
DEF7;5cm4;5cm6cm CAB8km10km6kmEDF3—m6—m3—m ABC4.5m5m7.5mDEF15m17m8m FEO5;8cm4;2cmBA3;9cm8;9cm 5;26;1CBDA8;93;9 E.8785 Consider the triangle DEF shown below with measures : DE = 7.5 cm ; EF = 6 cm ; DF = 4.5 cm Show that the triangle DEF is a right triangle with the vertex of the right angle specified. 4. Reciprocal and contrapositive of the Pythagorean theorem E.1063 Consider the two triangles ABC and DEF shown below freehand and whose exact side measures have been indicated : Specify whether these triangles ABC and DEF are rectan-gles. E.1064 Consider the two triangles ABC and DEF shown freehand below, with the exact side measures in-dicated : Determine whether or not these triangles are rectangles. E.1065 Determine the type of each of the triangles below : 5. Reciprocal and Pythagorean theorem E.1071 The configuration below is composed of the two triangles ABO and OEF where the latter is right-angled at F and where : EO = 5.8 cm ; OF = 4.2 cm ; OA = 3.9 cm OB = 8.9 cm ; AB = 2 × EF Determine the nature of the triangle ABO . E.1068 Consider the two triangles ABC and BCD shown below : 1 Calculate the length of segment [ BC ] rounded to the nearest tenth. 2 Is the triangle ABC a rectangle? https://chingmath.fr chapExoCorrec/8785 sacados/8785 DEF7;5cm4;5cm6cm chapExoCorrec/1063 sacados/1063 CAB8km10km6kmEDF3—m6—m3—m chapExoCorrec/1064 sacados/1064 ABC4.5m5m7.5mDEF15m17m8m chapExoCorrec/1065 sacados/1065 chapExoCorrec/1071 sacados/1071 FEO5;8cm4;2cmBA3;9cm8;9cm chapExoCorrec/1068 sacados/1068 5;26;1CBDA8;93;9
ABCD9cm12cm16cm20cm ABCDM ABCDE1;8cm2;4cm5cm3;2cm2;4cm ABCDE1;8cm2;4cm5cm3cm2;4cm 5;6mm4;2mmCBDA8;9mm3;9mm 4;2cm4cm3;1cm6;6cmABCD E.351 Consider the triangle BCD and its height [ AC ] . 1 Determine the length of segment [ BC ] . 2 Is the triangle BCD right-angled at C ? Hint: we recall the following perfect squares : 10 2 = 100 ; 14 2 = 196 ; 18 2 = 324 ; 22 2 = 484 11 2 = 121 ; 15 2 = 225 ; 19 2 = 361 ; 23 2 = 529 12 2 = 144 ; 16 2 = 256 ; 20 2 = 400 ; 24 2 = 576 13 2 = 169 ; 17 2 = 289 ; 21 2 = 441 ; 25 2 = 625 E.1070 Consider a rectangle ABCD having 20 cm in length and 15 cm in width. The point M is a point in the plane such that : AM =7 cm ; MC =24 cm A representation of this con-figuration is given opposite : 1 Justify that the triangle ADC is right-angled at D . 2 Determine the measure of segment [ AC ] . 3 Show that the triangle AMC is a right-angled triangle. E.5747 Consider the figure below où points A , B , C are aligned and the triangles ABE and EBD are rectangles at A and B , respectively. 1 Show that the segment [ BD ] has length 4 cm . 2 Justify that the triangle BCD is a right triangle in C . E.5750 Consider the figure below où points A , B , C are aligned and the triangles ABE and EBD are rectangles at A and B , respectively. 1 Show that the segment [ BD ] has length 4 cm . 2 Justify that the triangle BCD is not a right triangle. 6. Using approximate measurements. E.1795 Consider the two triangles ABC and BCD shown below ; the lengths of various segments have been noted on the figure. (Caution, drawing is not drawn at full size) 1 Calculate the length of segment [ BC ] . 2 Is the triangle ABC a rectangle? E.4533 The figure below shows two triangles ABC and ABD où the triangle ABD is rectangular at A . 1 Determine the measure of segment [ AB ] to the nearest millimeter. 2 Is the triangle ABC a right triangle? Justify your an-swer. https://chingmath.fr chapExoCorrec/351 sacados/351 ABCD9cm12cm16cm20cm chapExoCorrec/1070 sacados/1070 ABCDM chapExoCorrec/5747 sacados/5747 ABCDE1;8cm2;4cm5cm3;2cm2;4cm chapExoCorrec/5750 sacados/5750 ABCDE1;8cm2;4cm5cm3cm2;4cm chapExoCorrec/1795 sacados/1795 5;6mm4;2mmCBDA8;9mm3;9mm chapExoCorrec/4533 sacados/4533 4;2cm4cm3;1cm6;6cmABCD
ABCI ABCDM ABC ABCS12cm9cm8cm ABCS5cm13cm15cm ABCD8cm3;9cm8;9cm E.1067 Let ABC be a triangle and I a point on the segment [ AB ] such that : CI = 5 ; 6 cm ; IB = 3 ; 3 cm ; BC = 6 ; 5 cm ; AC = 13 ; 5 cm 1 Show that triangle CIB is a right triangle at I . 2 Give the length of [ AI ] , rounded to the nearest millime-ter. E.1069 In the figure below, the quadri-lateral is a square with a side length of 3 cm , and point M is located such that : AM =3 ; 6 cm ; MC =2 ; 24 cm 1 Prove that AC ≈ 4 ; 24 cm . 2 Is triangle AMC a right triangle? Explain your answer. E.5375 Consider the triangle ABC rectan-gle isosceles in C shown opposite où the segment [ AB ] has measure 9 cm . Determine the approximate value to the nearest millimeter of the seg-ment [ AC ] . 7. Using Approximate Measurements in Space E.2935 In space, consider the tetrahedron SABC . The following angles are right angles : ∠ SAB ; ∠ SBC ; ∠ SAC . We have the following measurements : SA = 12 cm ; AB = 9 cm ; BC = 8 cm 1 Show that the length of segment [ AC ] verifies : AC 2 = 145 cm . 2 Show that the triangle ABC is right-angled B . E.2934 In space, consider the tetrahedron SABC . The following angles are right angles : ∠ SAB ; ∠ SBC ; ∠ SAC We have the following measurements : AB = 5 cm ; SB = 13 cm ; SC = 15 cm 1 Determine the length of [ BC ] rounded to the millimeter. 2 a Determine the length of segment [ SA ] . b Deduce the value of AC . 3 Is the triangle ABC a rectangle? (use exact value of length BC ) . 8. Quadrilateral and reciprocal of the Pythagorean theorem E.4441 Consider the parallelogram ABCD shown below : https://chingmath.fr chapExoCorrec/1067 sacados/1067 ABCI chapExoCorrec/1069 sacados/1069 ABCDM chapExoCorrec/5375 sacados/5375 ABC chapExoCorrec/2935 sacados/2935 ABCS12cm9cm8cm chapExoCorrec/2934 sacados/2934 ABCS5cm13cm15cm chapExoCorrec/4441 sacados/4441 ABCD8cm3;9cm8;9cm
ABCDO10;5cm8;5cm6cm ABCH8;4;m11;2m14m ABCD Show that the parallelogram ABCD is a rectangle. E.4442 Consider the quadrilateral ABCD shown below : 1 Justify that ABCD is a parallelogram. is 2 ABCD a rectangle? Justify your answer. 3 ABCD is it a rhombus? Justify your answer. 9. Pythagorean theorem/reciprocal and areas E.4532 In the rainforest, a family of natives builds a hut of which a diagram is given below : 1 Justify that the triangle ABC is a right triangle. 2 a Determine the area of the ABC façade of this hut. b Deduce the measurement of the height [ BH ] of the hut. 3 Determine the measure of segment [ HC ] . E.6056 John Michael owns a field, represented by the triangle ABC below. He buys the adja-cent field, represented by the triangle ADC , from his neigh-bor. This results in a new field formed by the quadrilateral ABCD . John Michael knows that the perimeter of his field ABC is 154 meters and that BC =56 m . His neighbor informs him that the perimeter of the field ADC is 144 meters and that AC =65 m . In addition, he knows that : AD =16 m . 1 a Justify that the lengths AB and DC are equal to 33 m and 63 m respectively. b Calculate the perimeter of the field ABCD . 2 Show that the triangle ADC is right-angled D . We admit that the triangle ABC is right-angled at B . 3 Calculate the area of the field ABCD . 4 Jean-Michel wants to fence his field with wire mesh. He goes to his regular dealer and comes across the following ad : Grid: 0.85 e per meter. How much will he pay to fence his field? 10. Pythagorean theorem/reciprocal in space https://chingmath.fr chapExoCorrec/4442 sacados/4442 ABCDO10;5cm8;5cm6cm chapExoCorrec/4532 sacados/4532 ABCH8;4;m11;2m14m chapExoCorrec/6056 sacados/6056 ABCD
ABCDEFGHM24m7m12m9m ABCDEFGHM40m30m24m18m ABCDEFGHM240m238m120m288m ABC5x4x12 E.8694 Consider the right block ABCDEFGH shown below with the following measurements given : AB = 24 m ; BC = 7 m ; CG = 12 m The point M belongs to the diagonal [ EG ] of the rectangle EFGH and such that : EM = 9 m 1 a In the right triangle EMA rectangular in E , deter-mine the measure of the segment [ AM ] . b In the right triangle ABC rectangle in B , determine the measure of segment [ AC ] . c In the right triangle MGC rectangle in G , determine the measure of the segment [ MC ] . 2 Deduce that the triangle ACM is a rectangle at M . E.8782 Consider the right block ABCDEFGH shown below with the following measurements given : AB = 40 m ; BC = 30 m ; CG = 24 m The point M belongs to the diagonal [ EG ] of the rectangle EFGH and such that : EM =18 m 1 Establish that : MA = 30 m 2 Establish that : MG = 3 Deduce that the triangle ACM is right-angled M . E.8781 Consider the right block ABCDEFGH shown below with the following measurements given : AB = 240 m ; BC = 238 m ; CG = 120 m The point M belongs to the diagonal [ EG ] of the rectangle EFGH and such that : MG = 288 m Establish that the triangle ACM is right-angled at M . 11. Literal expressions E.1331 Consider the triangle ABC below where some of the triangle’s measures are expressed in terms of an indeterminate value noted x : Without justification, determine a value of x for which the triangle ABC is right-angled at C . https://chingmath.fr chapExoCorrec/8694 sacados/8694 ABCDEFGHM24m7m12m9m chapExoCorrec/8782 sacados/8782 ABCDEFGHM40m30m24m18m chapExoCorrec/8781 sacados/8781 ABCDEFGHM240m238m120m288m chapExoCorrec/1331 sacados/1331 ABC5x4x12
ABC2x3x7x ABCx15x ABCx222−2xx MN1mmABCD PEFGH1mm E.1327 Consider the ABC triangle shown below where the lengths are given in terms of the in-determinate x : 1 Justify that, for x =5 , the triangle ABC is a right-angled triangle at C . 2 Is the triangle ABC right-angled when x =4 E.1329 Consider the triangle ABC shown below where some of its lengths are expressed in terms of an indeterminate value noted x : 1 Show that for x =12 , the triangle ABC is a right-angled triangle at C . 2 Justify that triangle ABC is right-angled for x =3 . 3 Is the triangle ABC right-angled for x =8 ? E.1330 Consider the triangle ABC shown below where some of its lengths are expressed in terms of an indeterminate value noted x : 1 Show that for x =8 , the triangle ABC is a right-angled triangle at C . 2 Justify that triangle ABC is right-angled for x =6 . 3 Is the triangle ABC right-angled for x =7 . 12. Share E.8835 1 Consider a rectangle ABCD having the measure : AB = 24 mm ; AD = 16 mm We place the points M and n , belonging to the segments [ CD ] and [ AD ] , respectively, as shown below : a Show that : MN =15 mm b The following measurements are assumed : MB =20 mm ; BN = 25 mm What is the nature of the triangle MNB ? Justify. 2 Consider a rectangle EFGH with measure : EF = 16 mm ; EH = 6 mm Consider the point P such that HP = 8 mm . This figure is shown below : Justify that the triangle EFP is not a right triangle. 13. Unclassified exercises E.6312 ABC is a triangle such that : AB = 5 cm ; BC = 7.6 cm ; AC = 9.2 cm https://chingmath.fr chapExoCorrec/1327 sacados/1327 fichierPlus/1327/ ABC2x3x7x chapExoCorrec/1329 sacados/1329 ABCx15x chapExoCorrec/1330 sacados/1330 ABCx222−2xx chapExoCorrec/8835 sacados/8835 MN1mmABCD PEFGH1mm chapExoCorrec/6312 sacados/6312
ABCPPérimètredeABP;29PérimètredeBPC;09 GHIJK7;5cm4cm3;6cm7;6cm GHIJK5;6cm3;3cm4cm5;2cm 28;9cm13;6cm25;5cmABC 1 Draw this triangle at full size. is 2 ABC a right triangle? 3 Using software, we constructed this triangle, then : we placed a point P movable on the side [ AC ] ; triangles ABP and BPC were drawn ; we displayed the perimeter of these two triangles. a We move the point P onto the segment [ AC ] . Où should it be placed so that the distance BP is as small as possible? b We now place the point P at 5 cm from A . Which of the triangles ABP and BPC has the larger perimeter? c We move the point P back to the segment [ AC ] . O where must it be placed so that the two triangles ABP and BPC have the same perimeter? E.8784 Consider the figure shown below où the quadrilateral GHIJ is a rectangle. We give the following mea-surements : GH =7.5 cm ; HI =4 cm ; IK =3.6 cm ; GK =7.6 cm Is the triangle GKI a right triangle? Justify your answer. E.8786 Consider the figure shown below où the quadrilateral GHIJ is a rectangle. We give the following mea-surements : GH =5.6 cm ; HI =3.3 cm ; IK =4 cm ; GK =5.2 cm Is the triangle GKI a right triangle? Justify your answer. E.8985 Consider the triangle ABC shown below and whose measures are known : AB = 13 ; 6 ;cm ; AC = 28.9 cm ; BC = 25.5 cm Show that the height of the triangle ABC from the vertex C measures 12 cm . https://chingmath.fr ABCPPérimètredeABP;29PérimètredeBPC;09 chapExoCorrec/8784 sacados/8784 GHIJK7;5cm4cm3;6cm7;6cm chapExoCorrec/8786 sacados/8786 GHIJK5;6cm3;3cm4cm5;2cm chapExoCorrec/8985 sacados/8985 28;9cm13;6cm25;5cmABC