Grade 7 / Triangles 108 exercises (including 105 corrected)

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yAxBFEtCuDwG DAMERO TriangleisocèleTriangleéquilatéralTrianglerectangleTriangleisocèlerectangleTriangle plat ABCDM ABCDEFG ChingQuizz : 3 exercises available for Quizz assessment : 1. Reminders E.5606 The following four angles are shown in the figure below : ∠ yAx ; ∠ FBE ; ∠ tCu ; ∠ wDG Using your protractor, give the measure of each of these an-gles and complete the table : Angle ∠ yAx ∠ FBE ∠ tCu ∠ wDG Measurement (in degrees) E.11271 Consider the rectangle DAME and the triangle MER shown below : (Lengths are indicated in the figure.) Give the type of the triangles below : a EMA b EDO c ERA d MER e EOA Reminders: types of different special triangles : E.5607 The figure below represents a quadri-lateral ABCD . M is a point on segment [ AD ] : 1 Name, then measure, using the protractor, the four an-gles of the quadrilateral ABCD 2 Give the measure of the angle ∠ BMC E.10684 The figure below is composed of several triangles : Complete the dotted lines below with the symbols < , > , = in order to compare each pair of lengths : a AD + DB : : : AB b GD + DC : : : GC c FD + DA : : : AD + DC d AD + DB : : : EB + BD https://chingmath.fr chapExoCorrec/5606 sacados/5606 yAxBFEtCuDwG chapExoCorrec/11271 sacados/11271 DAMERO TriangleisocèleTriangleéquilatéralTrianglerectangleTriangleisocèlerectangleTriangle plat chapExoCorrec/5607 sacados/5607 ABCDM chapExoCorrec/10684 sacados/10684 ABCDEFG
ABCcab abcabcabc123 ABDE 2. Triangular Inequality E.1208 For each question, is it possible to construct a triangle with sides mea-suring a , b , and c ? Justify your answer. E.10685 Proposition: Triangular inequality In a triangle, the length of each side is less than or equal to the sum of the lengths of the other two sides. Construction of a triangle with 3 lengths: Three measurements can be used to draw a triangle if, and only if, the length of the largest measurement is less than or equal to the sum of the other two measurements. In each question, the measurements given can be used to con-struct a triangle. Give the triangle inequality that allows you to confirm the construction of the triangle: a a = 3 cm ; b = 8 cm ; c = 6 cm b d = 5 ; 4 cm ; e = 1 ; 8 cm ; f = 3 ; 8 cm E.1206 For each of the questions below, spec-ify whether triangle ABC is constructible or not, justifying your answer. a AB = 3 cm ; BC = 10 cm ; AC = 9 cm b AB = 5 cm ; BC = 3 cm ; AC = 1 cm c AB = 2 cm ; BC = 6 cm ; AC = 7 cm d AB = 2 cm ; BC = 1 cm ; AC = 1 ; 5 cm E.6501 For each question, specify if the triangle can be constructed and the nature of the triangle. Justify the answers. 1 AB = 5 cm ; BC = 7.5 cm ; AC = 4 cm 2 DE = 4 cm ; EF = 5 cm ; DF = 9 cm 3 GH = 6 cm ; HI = 2 cm ; GI = 4 cm 4 JK = 7 cm ; KL = 4 cm ; JL = 4 cm E.11435 For each of the questions below, specify whether triangle ABC is constructible or not, justify-ing your answer. a AB = 5 ; 2 cm ; BC = 7 ; 8 cm ; AC = 2 ; 6 cm b AB = 9 ; 8 cm ; BC = 3 ; 2 cm ; AC = 5 ; 7 cm c AB = 3 ; 8 cm ; BC = 5 ; 8 cm ; AC = 8 ; 1 cm E.10161 For each of the questions below, state whether or not the triangle ABC is constructible by justifying your answer. a AB = 3 cm ; BC = 7 cm ; AC = 2 cm b AB = 80 cm ; BC = 120 cm ; AC = 200 cm c AB = 2 cm ; BC = 3 cm ; AC = 3 cm E.1224 Is it possible to construct the trian-gle MNP such that : MN =6.7 cm ; MP =4.7 cm ; NP =11.5 cm Justify your answer. 3. Constructions of triangles with 3 measurements E.10686 In the box below, construct the two triangles : the triangle ABC such that : AB = 5 cm ; AC = 6 cm ; BC = 7 cm triangle DEF such that : DE = 4 cm ; DF = 8.5 cm ; EF = 6.5 cm 4. Triangular inequality: case of equality https://chingmath.fr chapExoCorrec/1208 sacados/1208 ABCcab abcabcabc123 chapExoCorrec/10685 sacados/10685 chapExoCorrec/1206 sacados/1206 chapExoCorrec/6501 sacados/6501 chapExoCorrec/11435 sacados/11435 chapExoCorrec/10161 sacados/10161 chapExoCorrec/1224 sacados/1224 chapExoCorrec/10686 sacados/10686 ABDE
OIOIOIOIOIOIOIOI 3cm1cm2cmABC3cm1cmFDEDEFest isocèle enF ABCDM E.5613 In the frame below, consider a part of a graduated line où the point O is the origin of the gradu-ated line and the point I is its unit. 1 Draw the circle C with center O and radius 5 7 . 2 Draw the circle C of center I and radius 2 7 . 3 How many points M in the plane verify in the plane the relations : OM = 5 7 ; IM = 2 7 What can we say about the position of the points O , I , M ? E.5614 Consider three points A , B and C aligned such that : AB = 4 ; AC = 7 cm ; BC = 3 cm Which of the statements below is true? a A ∈ BC b B ∈ AC c C ∈ AB E.10162 1 Consider the points D , E and F such that : DE = 9 cm ; DF = 6 cm ; EF = 4 cm Do the points D , E and F line up? 2 Consider the points G , H and I such that : GH = 5 cm ; GI = 12 cm ; HI = 7 cm Do the points G , H and I line up? E.1834 Consider an isosceles triangle MNP with principal vertex N . In each of the following cases, justify whether or not it is possible to draw this triangle: 1 PM =7 cm ; MN =3 cm 2 NM =3 cm ; PM = 6 cm 3 NM = 4 cm ; MP = 5 cm E.1207 For each of the triangles below, say why the information given on the drawing does not corre-spond to the representation made. Justify your statements. E.10687 Consider the quadrilateral ABCD shown below and the point M belonging to the diagonal [ BD ] : 1 Make a conjecture about the nature of the quadrilateral ABCD ? What observations are you basing your conjec-ture on? 2 Complete the dotted lines using the symbols < , > , or = : a AM : : : AD + DM b CB + DC : : : BD c AM + MC : : : AC d DM + BM : : : DB e DM : : : DB + MB E.10163 Consider the points J , K and L such that : JK = 6 cm ; JL = 4 cm ; KL = ? cm What should be the length of the segment : 1 so that the point L belongs to the segment [ JK ] ? 2 so that the point J belongs to the segment [ KL ] ? E.6502 In the two questions below, consider that the triangle ABC is a flat triangle and has the following measures : AC = 8 cm ; BC = 5 cm For each of the questions, determine the length of the segment [ AB ] that verifies the requested condition : 1 point B belongs to segment [ AC ] . 2 point C belongs to segment [ AB ] . E.10787 1 Consider the triangle ABC such that : AB = 5.3 cm ; AC = 3.8 cm Propose a measurement for segment [ BC ] so that trian-gle ABC is constructible, non-flat, and segment [ BC ] is the longest side of this triangle. 2 We consider the three measurements a , b , c : a = 8.2 cm ; b = 5.4 cm Propose a value for the measurement c such that it is impossible to construct a triangle with these three mea-surements and such that the measurement a remains the largest of these three measurements. 3 Consider the triangle DEF such that : DE = 3.9 cm ; DF = 2.8 cm Give the measurement of segment [ EF ] so that triangle DEF is constructible and flat, and such that point F belongs to segment [ DE ] . 4 Consider triangle GHI such that : GH = 4.2 cm ; GI = 6.4 cm Give the measurement of segment [ HI ] so that triangle GHI is constructible and flat, and such that point H belongs to segment [ GI ] . https://chingmath.fr chapExoCorrec/5613 sacados/5613 OIOIOIOIOIOIOIOI chapExoCorrec/5614 sacados/5614 chapExoCorrec/10162 sacados/10162 chapExoCorrec/1834 sacados/1834 chapExoCorrec/1207 sacados/1207 3cm1cm2cmABC3cm1cmFDEDEFest isocèle enF chapExoCorrec/10687 sacados/10687 ABCDM chapExoCorrec/10163 sacados/10163 chapExoCorrec/6502 sacados/6502 chapExoCorrec/10787 sacados/10787
AB6cmDE5;5cm ABDE ABDE ABC50o5;5cm45o120o4cm5cmGHI 5. Constructions of triangles E.6504 In the frame below, complete the figure to obtain triangles ABC and DEF such that : 1 AB = 6 cm ; BAC = 45 o ; ABC = 55 o 2 DE = 5 ; 5 cm ; EF = 6 cm ; DEF = 40 o E.10776 Consider the configuration below : 1 Draw the triangle ABC such as : AB = 6 cm ; BAC = 40 o ; ABC = 75 o 2 Draw the triangle DEF such that : DE = 4 cm ; EF = 5 cm ; DEF = 105 o E.11436 Consider the configuration below : 1 Draw triangle ABC such that : AB = 6 cm ; AC = 5 cm ; BC = 7 cm 2 Draw triangle DEF such that : DE = 5 cm ; EDF = 45 o ; DEF = 75 o E.1796 Construct the triangles below in real sizes : E.1209 Draw the following triangles accord-ing to the given measurements : 1 The triangle ABC such that : AB = 7 cm ; AC = 4 cm ; BC = 5 cm 2 triangle DEF such that : EF = 5 cm ; DEF = 75 o ; EFD = 60 o 3 triangle GHI such that : GI = 5 cm ; GH = 4.5 cm ; IGH = 50 o E.1211 Draw the triangle ABC verifying: AB = 7 cm ; BC = 9 cm ; ABC = 40 o E.1798 In each case, construct a triangle ABC such that : 1 AB = 7 cm ; BAC = 110 o ; ABC = 30 o 2 DFE = 55 o ; FE = 5 cm ; FD = 7 cm 3 GH = 8 cm ; GI = 10 cm ; HI = 6 cm https://chingmath.fr chapExoCorrec/6504 sacados/6504 AB6cmDE5;5cm chapExoCorrec/10776 sacados/10776 ABDE chapExoCorrec/11436 sacados/11436 ABDE chapExoCorrec/1796 sacados/1796 ABC50o5;5cm45o120o4cm5cmGHI chapExoCorrec/1209 sacados/1209 chapExoCorrec/1211 sacados/1211 chapExoCorrec/1798 sacados/1798
DEABC AB ABCDEF??GHI68o? 37o6;5cmDEF4;2cm72oJKL E.10165 In this question, we will see that it is possible to draw two triangles DEF verifying the condi-tions : DE = 8 cm ; DF = 7 cm ; ∠ FED = 50 o 1 Draw the segment [ DE ] and a half-line [ Ex ) verifying: ∠ DEx = 50 o 2 Place two points F and F  belonging to the half-line [ Ex ) and located at 7 cm from the point D . E.10164 Draw two triangles JKL verifying the measures : KL = 7 cm ; LJ = 5.5 cm ; ∠ LKJ = 50 o E.6 1 By measuring the lengths of its sides, reproduce triangle ABC . 2 Only part of triangle DEF has been shown above. Take the necessary measurements with your ruler and protractor to reproduce triangle DEF in its actual size. E.5722 Complete the figure below : Using the tracing program below : 1 Draw the triangle ABC such that : AB = 7 cm ; ∠ BAC = 75 o ; ∠ ABC = 50 o 2 Place the point M such that : M ∈ [ BC ] ; ∠ MAB = 35 o 3 Place the point N outside the triangle ABC such that : N ∈ [ AM ) ; ∠ CBN = 36 o E.1210 Construct four different triangles such that each of these triangles has the following three prop-erties : one of its sides 7 cm , one of its other sides measures 5 cm and it has an angle of 30 o . 6. Constructions of particular triangles E.6480 We consider the three particular tri-angles below : 1 Give the nature of each of these triangles, justifying your choice. 2 Name, then give the measure of each of the angles shown with a ˇ?ı question mark. E.10166 Construct in real size the triangles below : https://chingmath.fr chapExoCorrec/10165 sacados/10165 chapExoCorrec/10164 sacados/10164 chapExoCorrec/6 sacados/6 DEABC chapExoCorrec/5722 sacados/5722 AB chapExoCorrec/1210 sacados/1210 chapExoCorrec/6480 sacados/6480 ABCDEF??GHI68o? chapExoCorrec/10166 sacados/10166 37o6;5cmDEF4;2cm72oJKL
ABDE 70o7cm25oABCDE 40o7cm25oABCDE E.6489 Consider the two segments shown below : 1 Construct the isosceles triangle ABC at A such that : AC = 5 cm ; BAC = 40 o 2 Construct the isosceles triangle DEF at F such that : DE = 5 ; 5 cm ; EDF = 60 o E.5707 Draw the triangles below : 1 triangle ABC is isosceles at A and has measures : AC = 6 cm ; ∠ BAC = 40 o 2 triangle DEF is isosceles at F and has measures : DE = 5 cm ; ∠ FDE = 50 o E.10167 Draw the triangles below : 1 The triangle ABC is right-angled at A and has measures : AB = 9 cm ; AC = 3 cm 2 triangle DEF is right-angled at D and has measures : DE = 8 cm ; ∠ DEF = 30 o 3 triangle GHI is right-angled at H and has measures : GH = 7 cm ; GI = 8 cm E.6490 Draw the triangles below : 1 triangle ABC is isosceles at A and has measures : AC = 7 cm ; ∠ BAC = 44 o 2 triangle DEF is isosceles at F and has measures : DE = 6 cm ; ∠ FDE = 52 o E.6493 Below are three triangles : triangle ABC is isosceles at A ; triangle ACD is right-angled at C ; triangle ABE is equilateral. Measurements are shown in Figure: Reproduce this figure in real dimension. E.10775 Three triangles are shown below : triangle ABC is equilateral. triangle ACD is isosceles in C ; triangle ABE is right-angled at A ; Measurements are shown on figure : Reproduce this figure in true dimension. 7. Height and perpendicular bisector (reminders) E.10688 Definition: let A and B be two distinct points of the plane. The single straight line passing through the middle of seg-ment [ AB ] and perpendicular to line ( AB ) is called the perpendicular bisector of segment . In the plane, we have the two segments [ AB ] and [ CD ] whose representations are given below : https://chingmath.fr chapExoCorrec/6489 sacados/6489 ABDE chapExoCorrec/5707 sacados/5707 chapExoCorrec/10167 sacados/10167 chapExoCorrec/6490 sacados/6490 chapExoCorrec/6493 sacados/6493 70o7cm25oABCDE chapExoCorrec/10775 sacados/10775 40o7cm25oABCDE chapExoCorrec/10688 sacados/10688
ABCD ABCD ABCDEF ABCED 1 a Using the graduated ruler and square, draw the bi-sector of each of these two segments. b Name O the point of intersection of the two bisectors. 2 a Draw the circle with center O and radius [ OA ] . b What do you notice? E.10689 Using a straightedge and compass, draw the bisector of each of the segments below : E.10690 Reminders: In a triangle ABC , the height from vertex A is the line passing through point A and perpendicular to the opposite side [ BC ] . The area of a triangle is given by the formula : A = b × h 2 where b is the length of one base and h is the measure of the associated height. Consider the two triangles ABC and DEF shown below : 1 a Draw the height of triangle ABC from C . Label H as the foot of the height. b Measure the following lengths : AB = : : : ; CH = : : : c Determine the area of triangle ABC . 2 a Draw the height of triangle DEF from vertex F . Label the foot of this height I . b Measure the following lengths : DE = : : : ; FI = : : : c Determine the area of triangle DEF . E.11359 Consider the triangle ABC and the segment [ DE ] below : 1 Draw the perpendicular bisector of segment [ DE ] . 2 a Draw the height from A . b Determine the area of triangle ABC . 8. Areas of rectangular triangles https://chingmath.fr ABCD chapExoCorrec/10689 sacados/10689 ABCD chapExoCorrec/10690 sacados/10690 ABCDEF chapExoCorrec/11359 sacados/11359 ABCED
ABCD5cm3cm 6m3;2m6;8mABC8m6m10mDEF ABC6;8cm6cm 4cm7cm5cmABCDE ABCDE5cm12cm13cm ABCDMNO9cm25cm ABCDEFG9cm216cm2 E.9995 Below is shown the rectangle ABCD : 1 Determine the area of the rectangle ABCD . 2 Deduce the area of the triangle ACD rectangle in D . E.6456 Consider the two triangles ABC and DEF : Determine the areas of triangles ABC and DEF . E.11132 Consider the triangle ABC , which is a right triangle at B and has a perimeter of 16 cm : Determine the area of the triangle ABC . 9. Rectangles and additive methods E.1688 The figure opposite is com-posed of the square BCDE and a triangle AEB right-angled at E . 1 Calculate the perimeter of the figure. 2 Calculate the area of the figure. E.4227 The figure below is composed of a rectangle and a right triangle: 1 Determine the perimeter of the shaded figure. 2 Determine the area of the shaded figure. E.11131 Consider the figure below where : the quadrilateral ABCD is a square of side 5 cm ; the quadrilateral ABOM is a rectangle of 9 cm 2 area; triangle NOC is right-angled at O where N is the mid-point of segment [ MO ] . Determine the help of the shaded part (the polygon ABCNM ) . E.11134 The figure below is made up of the two squares ABCD and CEFG and the triangle BCE right-angled at E . Determine the total area of the figure. https://chingmath.fr chapExoCorrec/9995 sacados/9995 ABCD5cm3cm chapExoCorrec/6456 sacados/6456 6m3;2m6;8mABC8m6m10mDEF chapExoCorrec/11132 sacados/11132 ABC6;8cm6cm chapExoCorrec/1688 sacados/1688 4cm7cm5cmABCDE chapExoCorrec/4227 sacados/4227 ABCDE5cm12cm13cm chapExoCorrec/11131 sacados/11131 ABCDMNO9cm25cm chapExoCorrec/11134 sacados/11134 ABCDEFG9cm216cm2
ABCDMN2cm6cm4cm1cm 4cm2cm3cm6cm6cmABCDEFGH ABCDEF ABCDEF1;5cm3cm4cm ABCDNMOPRQST8cm4cm0;5cm2cm 10. Rectangles and subtractive methods E.11226 The shaded polygon ABCMN be-low : constructed from the rectangle ABCD and the points : M belonging to [ CD ] ; N belonging to [ AD ] . We have the measurements : AB =6 cm ; BC =2 cm ; CM =4 cm ; AN =1 cm 1 a Determine the length of segments [ DN ] and [ DM ] . b Determine the area of triangle DMN . 2 Determine the area of polygon ABCMN . E.1690 1 a Give the nature of polygons ABH and HGEF . b Give the area of each of these two polygons. 2 Calculate the area of polygon BCDEGH . E.1689 The figure shows the triangle AFE right-angled F . Point B is a point on segment [ AF ] and point D is a point on segment [ FE ] . The point C is such that the quadrilateral BCDF is a rect-angle. Here are some measurements on this figure : AB = 4 cm ; AF = 5.5 cm ; FD = 3 cm DE = 4 cm ; AE = 7.5 cm 1 Calculate the perimeter of the figure ˇgriséeı. 2 Calculate the area of figure ˇgriséeı. E.2634 Consider the figure opposite, representing a ABCD square with 4 cm sides. The point E belongs to the segment [ EB ] and the dis-tance from E to B measures 1.5 cm . The point F is a point on the figure verifying F ∈ [ AD ] FA = 3 cm 1 Determine the area of the right-angled triangle EBC . 2 Determine the area of triangle CDF . 3 Determine the area of the polygon AECF . E.11099 Consider the polygon P MNOPQRST shown below where ABCD is a rectangle and the right-angled triangles AMN , BOP , CQR , DST are iden-tical. Determine the area of the polygon P . Hint: remember to write out the steps of your reasoning. https://chingmath.fr chapExoCorrec/11226 sacados/11226 ABCDMN2cm6cm4cm1cm chapExoCorrec/1690 sacados/1690 4cm2cm3cm6cm6cmABCDEFGH chapExoCorrec/1689 sacados/1689 ABCDEF chapExoCorrec/2634 sacados/2634 ABCDEF1;5cm3cm4cm chapExoCorrec/11099 sacados/11099 ABCDNMOPRQST8cm4cm0;5cm2cm
ABCDEFG6cm2cm ABCDEF7cm3cm4;8cm1;6cm5cm 3;3cm5;6cm5;2cm3;9cm6;5cmABCDEF 12cm6cm4cmABCDE ADCB8m6m15m10m17mFHGE16m12m18m20m34m E.4228 The figure below consists of the two squares ABCD and EFGB : Determine the area of the shaded portion. E.11098 Consider the pentagon F shaded below : where the quadrilateral ABCD is a rectangle, E ∈ [ CD ] and F ∈ [ BC ] . 1 a Justify that ED =2.2 cm . b Determine the perimeter of the figure F 2 a Justify that CF =1.4 cm . b Determine the area of the figure F . E.11101 Consider the square ABCD below : Determine the area of the shaded part. 11. Introduction to the area of any triangles E.1696 We propose to calculate the area of the triangle in white. To do this, we’ll take the following steps : 1 a Calculate the area of the rectangle b Calculate the area of the two triangles ˇgrisésı ADE and BEC . c Deduce the area of the triangle ˇblancı. 2 By what calculation can we easily obtain the area of the triangle ABE using the numbers 6 and 12 . E.11089 Consider the two triangles ABC and EFG below : 1 a Determine the area of the triangle ABD . b Determine the area of triangle BCD . c Deduct the area of triangle ABC . 2 a Determine the area of the triangle EFH . b Determine the area of triangle EGH . c Deduct the area of triangle EFG . 3 Determine the values of the two quotients : AC × BD 2 ; FG × EH 2 12. Height and opposite side https://chingmath.fr chapExoCorrec/4228 sacados/4228 ABCDEFG6cm2cm chapExoCorrec/11098 sacados/11098 ABCDEF7cm3cm4;8cm1;6cm5cm chapExoCorrec/11101 sacados/11101 3;3cm5;6cm5;2cm3;9cm6;5cmABCDEF chapExoCorrec/1696 sacados/1696 12cm6cm4cmABCDE chapExoCorrec/11089 sacados/11089 ADCB8m6m15m10m17mFHGE16m12m18m20m34m
ABCDEF ABCDEF GHIJKL ABCDEF E.6702 Consider the two triangles ABC and CDE shown below : 1 In triangle ABC , draw the height from vertex B . 2 In triangle DEF , draw height from vertex D . E.1212 Consider the two triangles below : 1 In triangle ABC : a Name the side opposite vertex A . b Draw the height from vertex A . Name the foot of this height M . 2 In triangle DEF : a Label the side opposite vertex E . b Draw the height from vertex E . Name N the foot of this height. E.11100 Consider the two triangles below : 1 In triangle JKL : a Name the side opposite vertex J . b Draw the height from vertex J and name M the foot of this height. 2 In triangle GHI : a Name the side opposite vertex G . b Draw the height from vertex G and name N the foot of this height. 13. Areas of triangles E.10532 Consider the two triangles ABC and DEF shown below : 1 a Draw the height of triangle ABC from C . Name H the foot of the height. b Measure the following lengths : AB = : : : ; CH = : : : c Determine the area of triangle ABC . 2 a Draw the height of triangle DEF from vertex F . Name I the foot of this height. b Measure the following lengths : DE = : : : ; FI = : : : c Determine the area of triangle DEF . https://chingmath.fr chapExoCorrec/6702 sacados/6702 ABCDEF chapExoCorrec/1212 sacados/1212 ABCDEF chapExoCorrec/11100 sacados/11100 GHIJKL chapExoCorrec/10532 sacados/10532 ABCDEF
ABCDEF 8dm6dm10dm4;8dmABCH10m9m17m8mDEFI11cm30cm25cm8;8cmMNPJ ADCB12m16m5m20m13mEHGF40m9m21m41m50m ADCB12cm16cm5cm20cm13cmEHGF21cm20cm8cm29cm35cm 3cm4cmABCDIJ E.10531 Consider the two triangles ABC and DEF shown below : 1 a Draw the height of triangle ABC from C . Name H the foot of the height. b Measure the following lengths : AB = : : : ; CH = : : : c Determine the area of triangle ABC . 2 a Draw the height of triangle DEF from vertex F . Name I the foot of this height. b Measure the following lengths : DF = : : : ; EI = : : : c Determine the area of triangle DEF . E.7887 Determine the area defined by each of the triangles below : E.11090 In each case, determine the areas of the triangles ABC and EFG : E.11091 In each case, determine the area of the triangle ABC : 14. Areas of triangles: additive property of the area E.5587 Consider the quadrilateral ABCD shown below : I is the foot of the height from A in the triangle ABD . J is the foot of the height from C in the triangle BCD . We have the following measures : BD = 4 cm ; AI = 3 cm ; CJ = 4 cm Determine the area of the quadrilateral ABCD . https://chingmath.fr chapExoCorrec/10531 sacados/10531 ABCDEF chapExoCorrec/7887 sacados/7887 8dm6dm10dm4;8dmABCH10m9m17m8mDEFI11cm30cm25cm8;8cmMNPJ chapExoCorrec/11090 sacados/11090 ADCB12m16m5m20m13mEHGF40m9m21m41m50m chapExoCorrec/11091 sacados/11091 ADCB12cm16cm5cm20cm13cmEHGF21cm20cm8cm29cm35cm chapExoCorrec/5587 sacados/5587 3cm4cmABCDIJ
ABCDEFGHIJ4;5cm1cm2cm1cm2cm 3cmDA10cmDB ABCDOP5m2m 20m12m 15. Areas of triangles: figures composed by differences E.11133 Consider the shaded figure below, which is included in the square ABCD with side length 6 cm : 1 Determine the area of triangle HIJ . 2 a Without justification and in triangle EFG , give the length of the height from vertex F and the length of the side opposite vertex F . b Determine the area of triangle EFG . 3 Determine the area of the shaded figure. 16. Area of a disk E.9996 Below are shown : the half-disk D with center A and diameter 3 cm the quarter-disk D with center B and radius 10 m Determine the area of each of these figures rounded to the nearest tenth of a square centimeter. Note: we will use : ı ≈ 3 ; 1416 E.1694 The diagram below shows a table with a rectangular section and two semi-circular extensions. 1 Determine the perimeter of this table rounded to the nearest decimeter. 2 Determine the area of this table rounded to the nearest square meter. Note: we will use ı ≈ 3.14 17. Open problems E.5759 We have a triangular shaped bijour (represented below by the hatched triangle) and a rectangular gold leaf (represented by the grey rectangle below) : How many jewels can be covered with this gold leaf? Hint: several pieces of gold leaf can be used to cover the same piece of jewelry E.5760 A swimming pool has length 20 m and width 12 m A wooden walkway of width 2 m is laid out around the perimeter of the pool as shown in the sketch be-low : Determine the area in m 2 of this walkway. https://chingmath.fr chapExoCorrec/11133 sacados/11133 ABCDEFGHIJ4;5cm1cm2cm1cm2cm chapExoCorrec/9996 sacados/9996 3cmDA10cmDB chapExoCorrec/1694 sacados/1694 ABCDOP5m2m chapExoCorrec/5759 sacados/5759 chapExoCorrec/5760 sacados/5760 20m12m
ABCDOIM(d ABCMN ABCFDEGIC MNPABCIJ 18. Orthocenter E.572 Let A and B be two points in the plane, and let M be a point in the plane that does not lie on ( AB ) . Place point C such that M is the orthocenter of triangle ABC . E.1103 Let ABCD be a rectangle. The perpen-dicular bisector ( d ) of segment [ AC ] inter-cepts the line ( BC ) at M . 1 What do the straight lines ( AB ) and ( OM ) represent for the triangle AMC ? Justify? 2 What can you say about the straight lines ( CI ) and ( AM ) ? Justify. E.4966 Consider the triangle ABC shown be-low : The points M and N are the feet of the heights respectively originating from the vertices C and B . Using only an ungraduated ruler, place the foot P of the height arising from A . Justify your approach. E.1109 Consider the circle C with center F . We have : The points A , B and C belong to the circle C and are such that the triangle ABC is an isosceles triangle at A ; E is the foot of the triangle’s height ABC from the point A ; D is the middle of segment [ AB ] ; I is the point of intersection of the line ( AE ) and the line passing through B and perpendicular to ( AC ) ; The point G is the intersection of the straight line ( AE ) and the straight line ( CD ) . 1 a What is the orthocenter of the triangle ABC ? Jus-tify. b Draw the height of triangle ABC arising from C . 2 a Justify that point E is the midpoint of segment [ BC ] . b What is the center of gravity of the triangle ABC ? Justify your answer. c Place the middle of segment [ AC ] . 19. Center of Gravity E.1101 Consider a triangle ABC isosceles at B . Points I and J are the respective middles of segments [ AC ] and [ AB ] . M is the point of intersection of the straight lines ( BI ) https://chingmath.fr chapExoCorrec/572 sacados/572 chapExoCorrec/1103 sacados/1103 ABCDOIM(d chapExoCorrec/4966 sacados/4966 ABCMN chapExoCorrec/1109 sacados/1109 ABCFDEGIC chapExoCorrec/1101 sacados/1101 MNPABCIJ
(dABG ABCIJ ABCIJOKC ABCDOKI and ( CJ ) ; N is the point of intersection of the straight line ( BI ) and the straight line passing through C and perpendicular to the straight line ( AB ) ; P is the point of intersection of the line ( BI ) and the line passing through the point J and perpendicular to the line ( AB ) . Determine the position of the orthocenter, the center of grav-ity and the center of the circumscribed circle in the triangle ABC . E.1110 Find the point C such that the triangle ABC has the point G as its center of gravity. Propose a program for plotting the point C using the ungrad-uated ruler and compass. E.1108 Construction lines must remain visible on your sheet. 1 Perform the following drawing program : a Draw a line ( d ) ; b Plot two points C and I on the line ( d ) such that [ CI ] measures 6 cm . c Plot a point A not on the line ( d ) such that : IA = 4 cm . 2 Using a compass, draw point B such that [ CI ] is the median from C in triangle ABC . 3 a Recall the property: ˇ The center of mass of a triangle lies at 2 = 3 on each median, measured from the vertex. ı Mark the center of mass of the triangle ABC . b Using the center of gravity of the triangle ABC , locate points J and K , the respective midpoints of segments [ AC ] and [ BC ] . E.1106 Consider a triangle ABC . The points I and J are the respective middles of the sides [ AB ] and [ BC ] . Using only the ungraduated line, place the point K midpoint of the segment [ AC ] . Justify your approach. E.2930 In the plane, consider C a circle of center O and diameter [ AB ] ; C is a point in the plane such that the triangle ABC is right-angled at A ; the points I and J are the respective middles of the segments [ AC ] and [ BC ] ; the point K is the intersection of the straight lines ( IB ) and ( OC ) . Answer the following questions, justifying each of your an-swers : 1 Determine the center of the circumscribed circle of trian-gle ABC . 2 Determine the center of gravity of the triangle ABC . 3 Determine the perpendicular bisector to segment [ AB ] . E.1104 Let ABCD be a parallelogram and O the point of intersection of the diagonals. We note : I the middle of segment [ CD ] ; K the intersection of the straight lines ( OD ) and ( AI ) . Show that the straight line ( CK ) intersects the segment [ AD ] at its midpoint. 20. Unclassified exercises https://chingmath.fr chapExoCorrec/1110 sacados/1110 (dABG chapExoCorrec/1108 sacados/1108 chapExoCorrec/1106 sacados/1106 ABCIJ chapExoCorrec/2930 sacados/2930 ABCIJOKC chapExoCorrec/1104 sacados/1104 ABCDOKI
45oABC60o(d(dMNP8cm(AB==(MPACCB JesaisJ’utiliseJ’endéduis(det(d(d==(d JesaisJ’utiliseJ’endéduisDans le triangleABC:ABCoetBCAoLa somme de la mesure d’un angle dans un trian-gle vaut180o. np01234501111212131331414641515101051 E.5723 Consider the figure below : 1 Write the plotting program to obtain the figure below : 2 Plot this figure full-scale. E.393 Let A and B be two points of the plane such that AB =6 cm . Consider the circles C and C of respec-tive centers A and B of the same radius r . 1 What condition must the radius r of these two circles have in order for these two circles to intercept at two points? 2 It is assumed in this question that r =4 cm . Let’s note M and N these two points of intersection. Copy the table below and complete the deductive chain below, making it possible to show that the straight line ( MN ) is the perpendicular bisector of the segment [ AB ] . On sait On utilise On déduit AM = r BM = r N belongs to the mediatrix of [ AB ] E.396 1 Which proposition should be used in this deductive chain? 2 Which consequence is obtained in this chain? E.122 Here’s Pascal’s triangle for easy retrieval of a combinatorial value : This table represents the numbers n m where m represents the column number and n the line number (numbering start-ing at 0) . 1 a Circle the number 4 3 in green. Then, the number 4 4 . b Circle the number 5 4 in red. c What do we notice? To be able to build this table, you just need to use the fol-lowing three rules : For any natural number n , we have : n 0 =1 . Implied that there is only one set containing no num-bers : the empty set. For any natural number n , we have : n n =1 . For any natural number m and n such that m<n , we have : n − 1 m − 1 + n − 1 m = n m 2 Construct the next two rows of this table. E.1221 Draw the lines below, leaving traces of your constructions : 1 Draw a triangle EDF such that : ED = 8.5 cm ; FED = 60 o ; FDE = 30 o 2 Draw the height from F . 3 Draw the bisector of the angle FED using your ruler and compass. https://chingmath.fr chapExoCorrec/5723 sacados/5723 45oABC60o(d(dMNP8cm(AB==(MPACCB chapExoCorrec/393 sacados/393 chapExoCorrec/396 sacados/396 JesaisJ’utiliseJ’endéduis(det(d(d==(d JesaisJ’utiliseJ’endéduisDans le triangleABC:ABCoetBCAoLa somme de la mesure d’un angle dans un trian-gle vaut180o. sacados/122 np01234501111212131331414641515101051 chapExoCorrec/1221 sacados/1221
(dAB !40cm25cm!4cm20cm8cm ABCDEF GHIJKL E.1213 Let ( d ) be a straight line and a segment [ AB ] perpendicular to ( d ) but not intersecting the straight line ( d ) . 1 Reproduce an analogous figure on your copy 2 Place a point C such that ABC has ( d ) as the height from C . E.4226 A road sign is made up of two trian-gles (one inside, the other outside) which define a grey surface that will later be painted red. The panel has been shown twice, below, with its measure-ments indicated : Determine the area of the shaded part. E.1773 1 On the figure below, draw : The three bisectors of the triangle ABC . The three medians of the triangle DEF . The three medians of the triangle GHI . The three heights of the triangle LKJ 2 What property do each of these remarkable straight lines have? https://chingmath.fr chapExoCorrec/1213 sacados/1213 (dAB chapExoCorrec/4226 sacados/4226 !40cm25cm!4cm20cm8cm chapExoCorrec/1773 sacados/1773 ABCDEF GHIJKL
(d(d(d)ABC(D(D)DEF ABC(d(d ABC(d1(d2(d3(d4 E.1216 In each of the figures below, describe each of the lines drawn : E.1218 On the figure below, specify the nature of the straight lines ( d ) , ( d ) and (Δ) relative to the triangle ABC . E.1220 Determine the nature of each of the three straight lines below : E.5717 Consider an equilateral triangle ABC whose sides have measure 8 cm . The point D is placed in the plane so that : the triangle ABD is isosceles at B ; the half-right [ BC ) is the bisector of the angle ABD . 1 Show this figure in full size. 2 The following measure is assumed : BDA =30 o . Determine, justifying your approach, the measure of the other two angles of the triangle ABD . E.1217 Draw the lines below, leaving traces of your constructions : 1 Draw a triangle EDF such that : ED = 6 cm ; FED = 60 o ; FDE = 30 o 2 Draw the perpendicular bisector of segment [ ED ] and the perpendicular bisector of segment [ DF ] . 3 Note O the point of intersection of the two perpendicular bisectors. Draw the circumscribed circle of triangle EDF . E.1200 1 Give the definitions in a triangle GFE of the median drawn from G and the height relative to the side [ FG ] 2 Cite the triangular inequality in a triangle ABC . 3 Give the definition of the circumscribed circle of a trian-gle. What can be said about its center? https://chingmath.fr chapExoCorrec/1216 sacados/1216 (d(d(d)ABC(D(D)DEF chapExoCorrec/1218 sacados/1218 ABC(d(d chapExoCorrec/1220 sacados/1220 ABC(d1(d2(d3(d4 chapExoCorrec/5717 sacados/5717 chapExoCorrec/1217 sacados/1217 chapExoCorrec/1200 sacados/1200
ABCI ABCDEF ABCDEF ABCCentre de gravitéABCOrthocentreABCCentre du cercle circonscritABCCentre du cercle inscrit E.5628 Consider the triangle ABC shown below where I is the midpoint of segment [ AB ] . 1 Trace the height of triangle ABC originating from vertex C . 2 Justify that the triangles AIC and CIB are two triangles with the same area. E.1412 Consider a triangle ABC and the points D , E , F verifying: D is the middle of segment [ AE ] ; E is the middle of the segment [ BF ] ; F is the middle of segment [ CD ] . Such a figure is shown below : 1 a What is the name of the segment [ AF ] in the triangle ACD ? b Justify that segment [ AF ] divides triangle ADC into two triangles of equal measure. (note H the foot of the height from A ) . We’ll admit the following property: In a trian-gle, the median coming from a vertex divides this triangle into two triangles of equal length 2 a Justify that the three triangles EFD , EFC and EBC have the same area. b Deduce that triangle ABC has seven times the area of triangle DEF . E.1215 Draw the medians from the three ver-tices of each triangle in the figure below : What do you notice? E.1214 Construct a triangle ABC such that : height [ AH ] measure 6 cm , BAH =10 o its area measures 18 cm 2 E.1105 1 In each case, draw the remarkable straight lines required to obtain the requested point : 2 The point of intersection of the bisectors and the center of a circle particular to the triangle ABC . Specify the characteristics and properties of this circle. E.1861 1 Draw the triangle ABC such that : BAC = 35 o ; AB = 7 cm ; AC = 8 cm 2 Draw the perpendicular bisector of segment [ AB ] . 3 Draw the median from C . 4 Draw the height from C . E.1201 1 Give the definition of the perpendicular bisector. 2 Give the two properties related to the perpendicular bi-sector. https://chingmath.fr sacados/5628 ABCI sacados/1412 ABCDEF chapExoCorrec/1215 sacados/1215 ABCDEF chapExoCorrec/1214 sacados/1214 chapExoCorrec/1105 sacados/1105 ABCCentre de gravitéABCOrthocentreABCCentre du cercle circonscritABCCentre du cercle inscrit chapExoCorrec/1861 sacados/1861 chapExoCorrec/1201 sacados/1201
ABC E.6679 Consider the triangle ABC below : 1 Using the compass and a straightedge, draw the perpen-dicular bisector of segment [ AB ] . 2 Draw the median of triangle ABC from vertex C . 3 Using the square, trace the height of triangle ABC orig-inating from vertex B . https://chingmath.fr chapExoCorrec/6679 sacados/6679 ABC