Grade 6 / circles and medians 54 exercises (including 51 corrected)

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ABC ABCDEF ABCDEFGH ABCDEFGHIJ 1. Review: Plane Geometry E.11827 Notation: ( AB ) the line passing through points A and B . [ AB ) the half-line with origin A and passing through point B [ AB ] the segment with endpoints A and B Consider the figure below : For each question, circle the geometric object present in the above configuration : 1 [ AB ] ; ( AB ) ; [ AB ) ; [ BA ) 2 [ AC ] ; ( AC ) ; [ AC ) ; [ CA ) 3 [ BC ] ; ( BC ) ; [ BC ) ; [ CB ) E.11828 Link each sentence with the appropri-ate notation : The segment having as endpoints the points A and B ( AB ) The half-line with origin A and passing through the point B AB The distance separating the points A and B [ AB ] The straight line passing through the points A and B [ AB ) E.11829 We consider six points of plan repre-sented below : Copy and complete the dotted by the symbol corresponding among = ∈ and ∈ : a D : : : ( AE ) b A : : : [ EC ) c B : : : [ AE ] d C : : : [ FE ) e E : : : [ BD ] f B : : : [ AC ] E.11830 We consider the following configuration of several points in the plane : Copy and complete the blanks using the symbols ∈ , = ∈ , == and ⊥ . a G : : : : : : ( AH ) b ( BF ) : : : : : : ( AE ) c D : : : : : : [ EA ) d ( BH ) : : : : : : ( GC ) Hint: answers are checked for accuracy with the ruler and square. E.11831 On considère la figure ci-dessous : À l’aide du compas, déterminer tous les segments ayant la même longueur que les segments [ AB ] et [ CD ] . Indication : on les codera avec le symbole adéquat. 2. E.10784 Consider the following 26 points : https://chingmath.fr chapExoCorrec/11827 sacados/11827 ABC chapExoCorrec/11828 sacados/11828 chapExoCorrec/11829 sacados/11829 ABCDEF sacados/11830 ABCDEFGH sacados/11831 ABCDEFGHIJ chapExoCorrec/10784 sacados/10784
BCDEFGHIJKLMNOPQRSTUVWXYZA CABCDEFGHI PQABCDEFGHIJKLMNO EC1AC2FC3CC4BC5DC6 OABCDEC List all points that are three centimeters away from point A . E.2321 Consider the circle C of center O shown op-posite. Copy and complete the following state-ments using the signs ∈ and ∈ to indicate whether or not a point belongs to the circle: 1 A : : : C 2 B : : : C 3 C : : : C 4 D : : : C 5 E : : : C 6 F : : : C 7 G : : : C 8 O : : : C E.11832 Dans le plan, on considère les dix-sept points ci-dessous : 1 a Tracer le cercle C de centre P et passant par le point K . b Ce cercle met en évidence cinq points équidistants au point P . Compléter les pointillés suivants : P : : : = P : : : = P : : : = P : : : = P : : : = : : : cm 2 Un cercle C de centre Q permet de définir cinq point équidistants du point Q . Tracer ce cercle et compléter les pointillées: Q : : : = Q : : : = Q : : : = Q : : : = Q : : : = : : : cm E.6527 On the figure below are represented : six circles C 1 , C 2 , C 3 , C 4 , C 5 , and C 6 ; six points A , B , C , D , E and F of the plane. Associate each circle with its center. 3. Circle and definition E.1550 Consider the circle C drawn op-posite with center O . Label each of the segments shown in the figure, name them, and describe their nature. https://chingmath.fr BCDEFGHIJKLMNOPQRSTUVWXYZA chapExoCorrec/2321 sacados/2321 CABCDEFGHI chapExoCorrec/11832 sacados/11832 PQABCDEFGHIJKLMNO chapExoCorrec/6527 sacados/6527 EC1AC2FC3CC4BC5DC6 chapExoCorrec/1550 sacados/1550 OABCDEC
OACC ABCDEF ABCDEFC RayonDiamètreCordeADCFEABCFD ABCIJCC E.3780 Consider a circle C with center O and A , C two points on this circle: 1 Place point B on the circle such that [ AB ] is a diameter. 2 Using a compass and a ruler, draw the perpendicular to the line ( AB ) passing through point C . 3 a Draw the chord connecting points A and C in blue. b Draw the arc connecting points B and C in red. E.10855 In the plane, consider the following 6 points : 1 Draw circle C 1 with center A and radius segment [ AB ] . 2 Draw circle C 2 with diameter [ CD ] . 3 What is the circle with chord [ EF ] ? E.10785 Explain in a few words, in one sen-tence : 1 The difference between the radius and the diameter of a circle. 2 The difference between a circle and a disk. E.11833 Dans le plan, on considère le cer-cle C de centre O et cinq points de ce cercle: 1 Coder sur la figure l’ensemble des segments ayant la même longueur. 2 Cocher les cases correctes. E.11834 L’exercice n’existe pas. 4. Circle E.4079 Consider two circles C and C with centers I and J , respectively, and equal diameters. The points A , I , B , J , and C are aligned. 1 Justify that the segment [ AB ] is a diameter of the circle C . 2 Which of the following statements are correct? C is the circle with center I . C is a circle with center I . C is the circle with center J and diameter [ AB ] . C is the circle with center J and diameter AB . https://chingmath.fr chapExoCorrec/3780 sacados/3780 OACC chapExoCorrec/10855 sacados/10855 ABCDEF chapExoCorrec/10785 sacados/10785 chapExoCorrec/11833 sacados/11833 ABCDEFC RayonDiamètreCordeADCFEABCFD sacados/11834 chapExoCorrec/4079 sacados/4079 ABCIJCC
ABCCCD ABO ABCDEFGHIJKL ABCDEF ABC1IC2MC3 E.2834 Consider the figure below, which consists of circle C with center B and diameter [ AC ] and circle C with center A and radius [ AD ] . Circle C passes through point B . 1 List all segments of equal length in this figure, justifying your answers. 2 Compare the following lengths, giving reasons : a AC et AB + BC b DA + AC et DC Hint: To compare these lengths, you can use a compass to transfer the lengths to the line below : E.10817 ˇ Ying and Yang ı is a Taoist concept representing two op-posing but complementary forces. This figure is composed of a circle and two semicircles. To construct this figure (right) , we use the circle with center O such that points A and B belong to C and such that points O , A , B are aligned. Describe the circles and semicircles making up this figure. E.10816 The Fibonacci spiral makes it possible to construct a spiral by quarter circles built from squares reduced one from the other thanks to the golden ratio. The figure below shows part of the Fibonacci spiral with four quadrants. Describe each of these quarter circles. 5. Circle and equality of lengths E.2785 Consider the figure below : 1 Name all segments of the same length as [ EC ] . 2 Note a the length of segment [ FC ] . Name all the points on the figure belonging to the circle of center D and radius a . E.6542 Consider the figure below : The point I is the middle of the segment [ AB ] ; The circle C 1 has the point B as its center and passes through A ; The circle C 2 has center I and passes through the point A . The point M belongs to the circle C 1 and is such that the circle C 3 of center M passes through the points A and B 1 For each of the circles C 1 , C 2 , and C 3 , specify the nature of the segment [ AB ] . https://chingmath.fr chapExoCorrec/2834 sacados/2834 ABCCCD chapExoCorrec/10817 sacados/10817 ABO chapExoCorrec/10816 sacados/10816 ABCDEFGHIJKL chapExoCorrec/2785 sacados/2785 ABCDEF chapExoCorrec/6542 sacados/6542 ABC1IC2MC3
ABCD ABCDEMNC1C2C3 ABCDEHGF 2 Place the point C diametrically opposite the point A in the circle C 1 . 3 What feature does the triangle ABM have? Justify your answer. E.10317 Perform the following tracing pro-gram: 1 Place a point O on this line ( d ) and draw the circle C with center O and diameter 4 cm . 2 The circle C intercepts the line ( d ) at the points A and B . 3 Draw the circle C of center A . It intercepts the straight line ( d ) at a new point named M . 4 Draw the circle C with center B . It intercepts the straight line ( d ) at a new point named N . E.6211 In the plane, consider the four points A , B , C , D represented below : 1 Perform the following plot program : Draw the segment having for extremity the points A and C . Draw the half-line with origin A and passing through the point B . Draw the line passing through the points B and D . 2 Copy the program using mathematical notation. 3 a Draw the circle C with diameter [ AC ] . Note O its center. b Draw the circle C with center D and passing through the point B . c What length equalities can be observed in this figure? E.10314 Consider the figure below where : AB = BC = CD = DE = 2 cm C 1 , C 2 , C 3 are semicircles of respective diameters [ AC ] , [ BD ] , [ CE ] . 1 a Justify that BC = BM . b Justify that CB = CM = CN = CD . c Justify that DC = DN . 2 a What are the natures of the triangles BCM and CDN ? b We name I the middle of segment [ BC ] and J the middle of segment [ CD ] . What is the nature of the quadrilateral MNJI ? c Justify that the quadrilateral BCNM is a rhombus. 6. Circle and grid E.10815 In the plane below, consider the eight points shown below : Without using either the compass or the graduated ruler, an-swer the following questions : 1 Consider the circle C with center A and passing through the point B . a Justify that the point C belongs to the circle C . b Justify that the point D does not belong to the circle C . 2 Consider the circle C with center E and passing through the point B . Complete the dotted lines below with the symbols ∈ and ∈ . Justify your statements. B : : : C ; F : : : C ; G : : : C ; H : : : C Subsidiary question: points B , D , H belong to the same circle. Determine the center of this circle. 7. Construction program https://chingmath.fr chapExoCorrec/10317 sacados/10317 chapExoCorrec/6211 sacados/6211 ABCD chapExoCorrec/10314 sacados/10314 ABCDEMNC1C2C3 chapExoCorrec/10815 sacados/10815 ABCDEHGF
DBEFCA(AC==(BE(EF==(BDC(d AB(dCD E.10313 1 Carry out the construction program below : Draw a segment [ AB ] such that AB =5 cm Draw the circle C with center A and passing through the point B . Draw the circle C of center B and diameter 10 cm . Name M and N the two points of intersection of the circles C and C . Draw the line ( d ) passing through the points M and N . Name P the point of intersection of the line ( d ) and the segment [ AB ] . 2 a What can be said about the relative position of the straight lines ( AB ) and ( d ) ? b What can be said about the position of the point P on the segment [ AB ] ? 8. Write a construction program E.10312 Consider the configuration below : Write the contruction program starting with : ˇ Draw a triangle BED . ı 9. Introduction to mediators E.2819 Consider the two segments [ AB ] and [ CD ] shown below : 1 What is the name of the straight line ( d ) relative to the segment [ AB ] ? Justify your answer. 2 a Place two points M and N on the line ( d ) on either side of the line ( AB ) b Using your compass, compare the following two pairs of lengths : AM and BM ; AN and BN 3 a Of the 10 points shown in the second figure, three of these points verify the relationship : CM = DM Highlight these three points. b What special feature do these three points have? https://chingmath.fr chapExoCorrec/10313 sacados/10313 chapExoCorrec/10312 sacados/10312 DBEFCA(AC==(BE(EF==(BDC(d chapExoCorrec/2819 sacados/2819 AB(dCD
ABC ABC(d(d ABCDE(d7(d8(d6(d4(d2(d1(d3(d5 ABCDFE E.11835 Dans le plan, on considère le triangle ABC : 1 À l’aide du compas, vérifier que le triangle ABC est isocèle en C . Coder l’égalité de longueur mise en évi-dence. 2 a Placer le point I milieu du segment [ AB ] . Coder l’égalité de longueurs obtenues. b Tracer la droite ( d ) passant par le point I et perpen-diculaire à la droite ( AB ) . c Remarquer que le point C appartient à la droite ( d ) . 3 a Placer un point D tel que le triangle ABD soit isocèle en D . Indiquer l’égalité de longueurs créée. b Que remarque-t-on? 10. E.5609 Consider the triangle ABC shown below and three lines ( d ) , ( d ) and (Δ) with their properties shown on the figure : Which of these three lines is the perpendicular bisector of one of the sides of triangle ABC ? Justify your answer. E.2805 Consider the polygon ABCDE be-low and the various straight lines intercepting it shown below : Which of the eight straight lines shown in the figure can be the bisector of one of the five sides of the polygon ABCDE . 11. Drawing a perpendicular bisector with a square E.3670 Using a ruler and square, draw the bisector of each of the segments below : https://chingmath.fr chapExoCorrec/11835 sacados/11835 ABC chapExoCorrec/5609 sacados/5609 ABC(d(d chapExoCorrec/2805 sacados/2805 ABCDE(d7(d8(d6(d4(d2(d1(d3(d5 chapExoCorrec/3670 sacados/3670 ABCDFE
ABCD ABCD CDABEF 12. A little further with the mediators E.3699 Consider the two segments [ AB ] and [ CD ] below. Note: the lines will be drawn using a ruler and set square ; the construction lines must be visible on the sheet. 1 a Draw the perpendicular bisector of segment [ AB ] . b Draw the perpendicular bisector of segment [ CD ] . 2 Draw a circle C and a circle C such that : Circles C and C have the same center. Circle C passes through points A and B . Circle C passes through points C and D . Circles C and C , having the same center, are called concen-tric circles . 13. Drawing a perpendicular bisector with a compass E.6442 Using a straightedge and compass, draw the bisector of each of the segments below : E.2804 In the frame below and using a straightedge and compass, draw the bisectors of the segments [ AB ] , [ CD ] , [ EF ] . https://chingmath.fr chapExoCorrec/3699 sacados/3699 ABCD chapExoCorrec/6442 sacados/6442 https://chingamath.lan/gestion/edit/?e=5612&lang=fr ABCD chapExoCorrec/2804 sacados/2804 CDABEF
-4-3-2-1012345-2-112345xxyy A(d ABC E.5612 Consider the Cartesian coordinate system below : 1 Place the three points below in the reference frame : A ( − 3 ; 2) ; B (1 ; 4) ; C (4 ; − 1) 2 a Using the ungraduated ruler and compass, draw the bisector ( d ) of the segment [ BC ] . b Through which particular point does the line ( d ) pass? 3 a Using the ungraduated ruler and compass, draw the perpendicular bisector (Δ) of the segment [ AB ] . b Give the coordinates of the point of intersection of the line (Δ) with the y-axis. E.2336 Carry out the following layout pro-gram using only the compass and the straightedge : 1 Draw the perpendicular to the line ( d ) passing through the point A . We will name M the point of intersection of this line with ( d ) . 2 Draw the perpendicular to the line (Δ) passing through the point A . We’ll name N the point of intersection of this line with (Δ) . 3 Draw segment [ MN ] and its perpendicular bisector. E.6234 Consider the triangle ABC shown below : Perform the drawing program below : 1 a Using the unmarked ruler and compass, draw the perpendicular bisector (Δ) of segment [ BC ] . b Label I as the midpoint of segment [ BC ] . 2 a Using the unmarked ruler and the set square, draw the parallel line ( d ) to the line ( AB ) passing through point C . b Label J as the intersection point of lines ( d ) and (Δ) . 3 a Using the set square, draw line ( d ) perpendicular to line ( AB ) and passing through point J . b Label K as the intersection point of lines ( AB ) and ( d ) . 14. E.1203 Draw a circle C with center O and radius 6 cm . Place two points A and B on this circle. 1 What is the nature of the triangle OAB ? Justify your answer. 2 Justify that the point O belongs to the perpendicular of [ AB ] ? https://chingmath.fr chapExoCorrec/5612 sacados/5612 -4-3-2-1012345-2-112345xxyy chapExoCorrec/2336 sacados/2336 A(d chapExoCorrec/6234 sacados/6234 ABC chapExoCorrec/1203 sacados/1203
ABCDI ABCIJKL ABCDOC ABCCDEFG E.1199 1 Express all the properties coded on the figure opposite 2 What is the center of the cir-cumscribed circle of the trian-gle BCD ? Explain why. E.5646 We consider the configuration be-low : Determine, justifying your approach, the center of the circum-scribed circle. E.5608 Consider a circle C with center O où the segment [ AB ] is a diameter. The points C and D are two points on the circle C such that : AC = AD . 1 Justify that the point O belongs to the perpendicular bisector of segment [ CD ] . 2 Justify that the line ( AB ) is the perpendicular bisector of segment [ CD ] . 15. E.6219 1 a Draw the perpendicular bisector ( d ) of the segment [ AB ] . b Draw the perpendicular bisector ( d ) of the segment [ BC ] . c Name O the point of intersection of the straight lines ( d ) and ( d ) . is d point O a particular point on this figure? Justify your assertion. 2 a Draw the line (Δ) parallel to the line ( BC ) and pass-ing pr the point O . b Name M and N the two points of intersection of the line (Δ) with the circle C . is c segment [ MN ] a particular segment in this figure? Jus-tify your assertion. 3 a Draw the median ( D ) of the segment [ ED ] . b What does the line ( D ) represent for the segment [ GF ] ? https://chingmath.fr chapExoCorrec/1199 sacados/1199 ABCDI chapExoCorrec/5646 sacados/5646 ABCIJKL chapExoCorrec/5608 sacados/5608 ABCDOC chapExoCorrec/6219 sacados/6219 ABCCDEFG
ABCD ABC ABC E.5610 The figure below shows the trape-zoid ABCD 1 Using the graduated ruler and square, draw the bisector ( d ) of the segment [ AB ] . 2 What elements must be checked on the figure to affirm that the straight line ( d ) is also the perpendicular bisec-tor of the segment [ CD ] ? 3 What does the line ( d ) represent for the trapezoid ABCD ? 16. E.6492 Consider the three points A , B and C shown below : 1 Using the graduated ruler and square, draw the bisectors of the segments [ AB ] and [ BC ] . 2 a Name M the point of intersection of these two bisec-tors. b Draw the circle C with center M and passing through the point A . What do we notice? 17. E.10366 Consider the triangle ABC below : 1 Using the ungraduated ruler and compass, draw the three https://chingmath.fr chapExoCorrec/5610 sacados/5610 ABCD chapExoCorrec/6492 sacados/6492 ABC chapExoCorrec/10366 sacados/10366 ABC
ABCI ABC ABC ABC bisectors of segment ABC . 2 Draw the circumscribed circle of the triangle ABC . E.6236 Consider the triangle ABC shown below où the point I is the midpoint of the segment [ AB ] . 1 Using the square, draw the perpendicular to the line ( AB ) passing through the point I . 2 Using the ungraduated ruler and compass, draw the per-pendicular bisector of segment [ BC ] . 3 Draw the circumscribed circle of the triangle ABC . E.6503 Consider the three points A , B and C shown below : 1 Using the ungraduated ruler and compass, draw the bi-sectors of segments [ AB ] and [ BC ] . (We’ll leave the con-struction strokes present) . 2 a Name M the point of intersection of these two bisec-tors. b Draw the circle C with center M and passing through the point A . What do we notice? E.2820 Consider the triangle ABC below : 1 Using a straightedge and compass, draw the perpendicu-lar bisectors of the three sides of the triangle ABC below : 2 Draw the circumscribed circle of triangle ABC . E.10391 Consider the triangle ABC below : 1 Using the ungraduated ruler and compass, draw the three bisectors of segment ABC . 2 Draw the circumscribed circle of the triangle ABC . E.1219 1 Draw a triangle ABC such that : AB = 6 cm ; AC = 7 cm ; BC = 5 cm 2 Use a ruler and compass to draw the three perpendicular lines of the ABC triangle. 3 Name O the intersection of the perpendicular bisectors. Draw the circumscribed circle of the triangle ABC https://chingmath.fr chapExoCorrec/6236 sacados/6236 ABCI chapExoCorrec/6503 sacados/6503 ABC chapExoCorrec/2820 sacados/2820 ABC chapExoCorrec/10391 sacados/10391 ABC chapExoCorrec/1219 sacados/1219
ABCD ABC E.3700 Consider the two segments [ AB ] and [ CD ] below. Tracings must be made with compass and ungraduated ruler ; they must be present on the sheet. 1 a Draw the perpendicular bisector of segment [ AB ] . b Draw the perpendicular bisector of segment [ CD ] . 2 a Draw the same circle C passing through points A , B , C and D . b Justify the position of the center of the circle C . The four points A , B , C and D belong to the same circle: these points are said to be cocycliques . 18. E.2315 Consider the triangle ABC below : 1 Draw the perpendicular bisectors of the following three segments : [ AB ] ; [ AC ] ; [ BC ] 2 What do we notice? We name I the point of intersection of the perpendicular bi-sectors. 3 Draw the circle with center I having radius IA distance. 4 What do you notice? Proposition-definition: any triangle (non-aplati) ABC admits a single circle that passes through its three vertices A , B and C . This circle is called the circumscribed circle of the tri-angle ABC and its center is the point of intersection of the perpendicular bisectors of the triangle’s three sides ABC . E.2324 1 Draw the triangle ABC whose side measures are: AB =5 cm ; AC =8 cm ; BC =7 cm 2 a Using the graduated ruler, find the middles of seg-ments [ AB ] and [ BC ] . b Draw the bisectors of the sides [ AB ] and [ BC ] . 3 Name O the point of intersection of the medians, then draw the circle with center O and radius [ OA ] . https://chingmath.fr chapExoCorrec/3700 sacados/3700 ABCD chapExoCorrec/2315 sacados/2315 ABC chapExoCorrec/2324 sacados/2324