Grade 6 / triangles 36 exercises (100% corrected)

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ABCcôtéopposéausommetCDEFsommetopposéaucôtéDE ABCDEF ABCDOMN ABCDEF ChingQuizz : 9 exercises available for Quizz assessment : 1. General information about triangles E.2403 Definition: Consider the two triangles ABC and DEF below : 1 Name the vertex opposite the [ BC ] side in the ABC tri-angle. 2 Name the side opposite the vertex E in the triangle DEF . 3 Name the side opposite the vertex B in the triangle ABC . 4 Name the vertex opposite side [ DE ] . E.10330 1 Let MNC be a triangle, what is the vertex opposite side [ MC ] ? 2 Let JKL be a triangle, which side is opposite the vertex K ? E.2405 Consider a square ABCD . Let O be the midpoint of the diagonal [ AC ] . We draw the perpendicular to [ DC ] passing through O ; it intercepts [ DC ] at M and [ AB ] at N . We trace The segment [ OB ] . 1 Determine the set of triangles defined by this figure. 2 a Name the side opposite A in the triangle ABC . b Name the side opposite A in the triangle ANO . 3 a Name the vertex opposite [ AB ] in the triangle ABO . b Name the vertex opposite [ AB ] in the triangle ABC . E.10331 Consider the figure below composed of 30 segments. How many triangles can be constructed in to-tal using the segments in this figure? 2. Rectangles E.10334 Definition: A triangle is said to be rectangle if one of its angles is a right angle. In a right-angled triangle, the side opposite the right angle is called the hypothenuse of the rectangle. Consider the two rectangular triangles below : Complete the following sentences : The ABC triangle is . . . . . . . . . because it admits a right angle at its vertex . . . . . . . Its hypothenuse is its side . . . . . . https://chingmath.fr chapExoCorrec/2403 sacados/2403 ABCcôtéopposéausommetCDEFsommetopposéaucôtéDE ABCDEF chapExoCorrec/10330 sacados/10330 chapExoCorrec/2405 sacados/2405 ABCDOMN chapExoCorrec/10331 sacados/10331 chapExoCorrec/10334 sacados/10334 ABCDEF
ABC ABCDEFGHIJKL IJKM DEFMNO The EDF triangle is rectangular at . . . . . . and admits . . . . . . as its hypothenuse. E.10912 Below is a right triangle ABC with C : 1 Name the side opposite vertex B . 2 Name the vertex opposite side [ BC ] . 3 What does side [ AB ] represent in triangle ABC ? E.10332 Consider the 4 triangles shown be-low : Which of these triangles are right-angled triangles? We will then specify the vertex of the right angle. Hint: we will use the square to verify the existence of a right angle. E.10336 1 Let MNP be a right triangle in M . Name its hy-pothenuse? 2 Let AXP be a triangle admitting a right angle at X . Name its hypothenuse? 3 Let IJK be a right triangle whose hypothenuse is side [ KI ] . Name the vertex of the right angle? E.10340 We consider the configuration be-low : obtained by the plotting program : draw a triangle IJK right-angled K draw the line ( d ) through the point K and perpendicular to the line ( IJ ) . Name M the point of intersection of the lines ( d ) and ( IJ ) . 1 Name the three right triangles present in this figure and their right angle vertex. 2 Complete each of the sentences below : a side [ IJ ] is the hypothenuse of the triangle . . . . . . rect-angle in . . . b The side [ JK ] is the hypothenuse of the triangle . . . . . . rectangle in . . . c side [ IK ] is the hypothenuse of the triangle . . . . . . rect-angle in . . . 3. Isosceles triangles E.10335 Definition: A triangle is said to be isosceles if any two of its sides have the same measure. In an isosceles triangle, the vertex common to dex sides of the same measure is called the principal vertex. In an isosceles triangle, the side opposite the principal vertex is called the principal base. Consider the two isosceles triangles below : Complete the following sentences : The DEF triangle is . . . . . . . . . because its two sides . . . . . . and . . . . . . have the same measure. The . . . . . . side is its principal base. Noting that . . . . . . = . . . . . . , we know that the triangle OMN is isosceles and its principal vertex is . . . . . . https://chingmath.fr chapExoCorrec/10912 sacados/10912 ABC chapExoCorrec/10332 sacados/10332 ABCDEFGHIJKL chapExoCorrec/10336 sacados/10336 chapExoCorrec/10340 sacados/10340 IJKM chapExoCorrec/10335 sacados/10335 DEFMNO
ABCDEFGHIJKLMNO ABCDEF ABCDE ABCDEFGHI ABCD E.10333 Consider the five triangles below : Which of these triangles are isosceles triangles? Specify their principal vertex. Hint: use the compass to check the equality of segment measures E.10337 1 Let MNP be an isosceles triangle in M . Which side is the major base of the MNP triangle? 2 Let ADP be a triangle such that PA = PD . What is the principal vertex of the triangle ADP .? 3 Let UCJ be an isosceles triangle admitting side [ CJ ] as its major side. What is the principal vertex of this trian- gle? E.2323 Name the set of isosceles triangles represented in the figure opposite E.1547 1 Name all apparent isosceles triangles in the figure oppo-site. 2 Two isosceles triangles have not been drawn in this figure ; which ones? 4. Particular triangles E.10338 We consider the three triangles be-low : Among these three triangles, only one is a right isosceles tri-angle. Which one is it? Hint: to determine the correct triangle, use your geometry tools (square and compass) . E.10339 Definition: a triangle is said to be equilateral if its three sides all have the same measure. Consider the square ABCD shown below : 1 Draw the triangle ABE equilateral located inside the square ABCD . 2 Draw the triangle BCF equilateral located outside the square ABCD . https://chingmath.fr chapExoCorrec/10333 sacados/10333 ABCDEFGHIJKLMNO chapExoCorrec/10337 sacados/10337 chapExoCorrec/2323 sacados/2323 ABCDEF chapExoCorrec/1547 sacados/1547 ABCDE chapExoCorrec/10338 sacados/10338 ABCDEFGHI chapExoCorrec/10339 sacados/10339 ABCD
F(ECI(ABIFABCDE ABC1D1C2D2C3D3C4D4C5D5C6D6C7D7C AB E.10913 Consider the rectangle ABCD be-low : 1 How many traced triangles does this figure have? 2 Give the nature of each of these triangles. 5. Drawing triangles E.347 The drawing below represents the seg-ment [ AB ] such that : AB = 6 cm The circles C 1 , C 2 ,. . . , C 5 are circles with center A and radius 1 cm , 2 cm ,. . . , 5 cm respectively. Similarly, circles D 1 ,. . . , D 5 are circles of center B and radius from 1 cm to 5 cm : 1 Explain why the triangle ABC has the following mea-sures : AB =6 cm ; AC =5 cm ; BC =4 cm 2 On the graph above ; specify the position of the point D such that the triangle ABD has dimensions : AB =6 cm ; AD =3 cm ; BD =5 cm 3 a What can you say about a triangle ABE whose di-mensions verify: AB = AE + EB ? b Give an example. E.10341 1 Draw the triangle JKL having dimensions : JK =8 cm ; KL =7 cm ; JL =6 cm 2 Draw the triangle MNO having dimensions : MO =10 cm ; NO =5 cm ; MN =6 cm E.10914 Below, consider the segment [ AB ] such that : AB = 4 cm 1 In the shaded part of the plane, construct the triangle ABC such that : AB = 4 cm ; AC = 5 cm ; BC = 4 cm 2 In the white part of the plane, construct the triangle ABD such that : AB = 4 cm ; AD = 3.5 cm ; BD = 6 cm E.2866 1 a Draw triangle ABC with the following dimensions : AB =6 cm ; BC =6 cm ; AC =6 cm b Draw triangle DEF with the following dimensions : DE =5 cm ; DF =7 cm ; EF =7 cm c Draw triangle GHI with the following dimensions : HI =5 cm ; GI =3 cm ; GH =4 cm 2 Describe the nature of each of these triangles. E.1548 Leave the construction lines on your figure. 1 Construct a triangle ABC such that : AB = 7 cm ; BC = 6 cm ; AC = 5 cm 2 Place the points E , F and G such that the triangles ABE , BCF and CAG are equilateral triangles positioned out-side the triangle ABC . 6. Drawing of polygons https://chingmath.fr chapExoCorrec/10913 sacados/10913 F(ECI(ABIFABCDE chapExoCorrec/347 sacados/347 ABC1D1C2D2C3D3C4D4C5D5C6D6C7D7C chapExoCorrec/10341 sacados/10341 chapExoCorrec/10914 sacados/10914 AB chapExoCorrec/2866 sacados/2866 chapExoCorrec/1548 sacados/1548
ABC7cm5cm8cmD5cm6cm ABC7cm4cm6cmD3cm6cmE3cm5cm 6cm7cm5cm7cmABCDEF ABCC E.10390 Reproduce the following figure us-ing a ruler and a compass : E.6319 Reproduce the following figure using a ruler and a compass : E.6333 The figure below is composed of 4 triangles where : measurements are plotted on the figures ; points F , C and D are aligned. Reproduce, full size, this figure. 7. Carry out a construction program E.2605 1 Perform the following plot program : a Draw a triangle ABC isosceles at A such that : AB = 4.5 cm ; BC = 3 cm b Draw the perpendicular bisector of segment [ BC ] using a ruler and compass. Note I the midpoint of segment [ BC ] . c Place a point J on the half-line [ IA ) . d Draw the quadrilateral BACJ . 2 What is the nature of the quadrilateral BACJ ? Justify your answer. E.2887 Consider a circle C and three points A , B , C on this circle, as shown below : 1 a Using a compass and an unmarked ruler, draw the perpendicular bisector ( d ) of the segment [ AC ] . https://chingmath.fr chapExoCorrec/10390 sacados/10390 ABC7cm5cm8cmD5cm6cm chapExoCorrec/6319 sacados/6319 ABC7cm4cm6cmD3cm6cmE3cm5cm chapExoCorrec/6333 sacados/6333 6cm7cm5cm7cmABCDEF chapExoCorrec/2605 sacados/2605 chapExoCorrec/2887 sacados/2887 ABCC
MN(dABCDI ABCDExxAMNOPQRyyM ABCDEFG b Label I as the point of intersection of ( d ) with the small arc AC . Label J as the point of intersection of ( d ) and ( AC ) . c Place point K such that : K ∈ ( d ) and JK = JI . d What is the nature of quadrilateral AKCI ? 2 We want to place points D and E on this figure so that quadrilateral ADBE is a square : a What can we say about lines ( DE ) and ( AB ) ? Justify your answers. b What can be said about segments [ BA ] and [ DE ] ? Jus-tify your answers. c Using a compass, draw the perpendicular bisector of segment [ AB ] . d Draw square ADBE . E.115 Consider the configuration where ABCD is a rectangle below : Copy and complete the dotted lines : 1 Draw a rectangle named . . . . . . 2 Draw segment [ : : : ] 3 Draw the perpendicular bisector ( : : : ) of the segment [ : : : ] . 4 The right ( : : : ) intercepts the [ : : : ] side at : : : and inter-cepts the [ : : : ] side at : : : . Then finalize the plot program for this figure. 8. E.11836 On considère les deux lignes brisées suivantes : 1 Laquelle de ces deux lignes brisées vous semble la plus grande? 2 À l’aide du compas, reporter la ligne brisée de A à E sur la droite ( xx ) ; faire de même pour la ligne de M à R sur la droite ( yy ) . 3 Comparer maintenant la longueur de ces deux lignes brisées. E.11837 La figure ci-dessous est composée de plusieurs triangles : 1 Compléter les pointillés ci-dessous avec les signes < , > , = afin de comparer chaque couple de longueur : a AB : : : CD b AB : : : AD c CE : : : DB d GB : : : FC e FC : : : AE 2 Faire de même : a FD + DA : : : AD + DC b AD + DB : : : EB + BD 3 Faire de même : a AD + DB : : : AB b GD + DC : : : GC c CD + DF : : : FC https://chingmath.fr chapExoCorrec/115 sacados/115 MN(dABCDI chapExoCorrec/11836 sacados/11836 ABCDExxAMNOPQRyyM chapExoCorrec/11837 sacados/11837 ABCDEFG
ABCDMN CCABCD (d(dABCIJKL(BC==(dAJJB E.11838 On considère le rectangle ABCD représenté ci-dessous ; un point M contenu à l’intérieur du rectangle; N est un point du segment [ DC ] . 1 Des codages sont présents sur la figure, préciser leurs significations. 2 a Quelle particularité possède le triangle AMD ? b Comparer les deux triangles ABM et CDM . 3 Comparer les longueurs suivantes à l’aide des symboles : = ; < ; > a DA : : : DM + MA b DC : : : DM + MC c DC : : : DN + NC d DN : : : DC + CN 9. Unclassified exercises E.2833 Let C be a circle of center A and C a circle of center B ; the circle C passes through the point B and the circle C passes through the point A . Note D and C the points of intersection of these two circles: 1 From quel (s) cercle (s) , is the segment [ AB ] the radius? 2 a Using the fact that the point A is the center of the circle C , justify the equality: AB = AC . b Similarly, justify that : BA = BC c Deduce that the triangle ABC has its three sides of equal measure. 3 a Justify that the point C is a point on the perpendic-ular bisector of the segment [ AB ] . b On this figure, what is the position of the perpendicu-lar bisector of segment [ AB ] ? Justify your statement. E.6317 We consider the following configura-tion : 1 Write the plot program for this configuration following the following two instructions : The plot program will be started with the sentence : ˇ Draw a triangle ABC having a right angle at C ı The word ˇ median ı will be used once in the plot pro-gram. 2 a Name the theorem for stating that the lines ( IK ) and ( CL ) are parallel. b Name the theorem for stating that the lines ( LK ) and ( AC ) are perpendicular. https://chingmath.fr chapExoCorrec/11838 sacados/11838 ABCDMN chapExoCorrec/2833 sacados/2833 CCABCD chapExoCorrec/6317 sacados/6317 (d(dABCIJKL(BC==(dAJJB
ABCI JesaisJ’utiliseJ’endéduisCACBCappartient à la médiatrice deAB. JesaisJ’utiliseJ’endéduisLa médiatrice d’un segment est la droite perpen-diculaire à ce segment et passant par le milieu dece segment. JesaisJ’utiliseJ’endéduisCetIsont deux points de la médiatrice deABLa médiatrice d’un segment est une droite et pardeux points, il passe une unique droite E.6179 The figure opposite shows a trian-gle ABC verifying the equality of lengths : AC = CB The point I is the middle of the segment [ AB ] . Complete the following deductive links: E.392 1 Setting up the figure : a Place 3 points O , A , B in the plane. b Draw the half-lines [ OA ) and [ OB ) . c Draw a circle with center O radius 5 cm . d Name M and N the points of intersection of the circle C with the half-lines [ OA ) and [ OB ) respectively. e Draw two circles C 1 and C 2 of radius 5 cm and centers M and N respectively. These two circles intercept at O and P . f Draw the quadrilateral OMPN . 2 Justify that the quadrilateral OMPN is a rhombus. E.1202 Quote the definition of the circumscribed circle and la (les) propriété (s) liée (s) to the circumscribed cir-cle. https://chingmath.fr chapExoCorrec/6179 sacados/6179 ABCI JesaisJ’utiliseJ’endéduisCACBCappartient à la médiatrice deAB. JesaisJ’utiliseJ’endéduisLa médiatrice d’un segment est la droite perpen-diculaire à ce segment et passant par le milieu dece segment. JesaisJ’utiliseJ’endéduisCetIsont deux points de la médiatrice deABLa médiatrice d’un segment est une droite et pardeux points, il passe une unique droite chapExoCorrec/392 sacados/392 chapExoCorrec/1202 sacados/1202