Outside the high school program / Space geometry 25 exercises (including 24 corrected)

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ABCDE ABCFig1ABCFig2 ABCDEFGH 1. Equestrian perspectives E.575 Here are the rules for representing a solid in cavalier perspective : Front view figures remain unchanged. Parallel straight lines are represented by parallel straight lines. The alignment of points is preserved. The length ratio is preserved : in particular a midpoint remains a midpoint. Hidden parts are shown as dotted lines. Here are incomplete representations of a cube in cavalier per-spective. Draw the missing lines, respecting the cavalier per-spective. a b c d E.3116 Consider the rectangular-based pyramid ABCDE shown below : 1 a Place the center of the rectangle ABCD on the fig-ure. b What property of cavalier perspective has been used to place the point O ? 2 a Place the center of gravity G of the triangle EBC . b What properties of cavalier perspective were used to draw this center of gravity? E.4947 Consider a quarter circle AC of a circle of centroid B represented in the plane (Fig. 1) and in space (Fig. 2) : In this exercise, we’ll use only the ungraduated ruler and the compass. 1 In figure 1 : a Place the middles of segments [ BA ] and [ BC ] . Justify your construction. b Place the midpoint of the arc AC . Justify your con-struction. 2 In figure 2 : a Place the middles of segments [ BA ] and [ BC ] . b Can we place the midpoint of the arc AC ? E.4941 Consider the ABCDEFGH cube shown below. 1 Determine the measure of the angle FGB . 2 a Give the nature of the triangle BEG . Justify your answer. b Give the measure of the angle EBG . https://chingmath.fr chapExoCorrec/575 sacados/575 chapExoCorrec/3116 sacados/3116 ABCDE chapExoCorrec/4947 sacados/4947 ABCFig1ABCFig2 chapExoCorrec/4941 sacados/4941 ABCDEFGH
ABCD MABCDEFGH ABCDMNP E.4943 In space, consider the regular tetrahe-dron ABCD with side 5 cm : all its faces are equilateral tri-angles and the foot of the height from a vertex is the center of the opposite face of that vertex. 1 In triangle ABC : a Draw the height from vertex C . Note I the foot of this height. Justify your construction. b Place the point G center of gravity of the triangle ABC . Justify. It is assumed that all the heights of an equilateral triangle with side a have measure a 3 2 : 2 Give the measure of segment [ CI ] . 3 a Give the measure of the segment [ CG ] . Justify your answer. b Determine the measure of segment [ DG ] . 4 Determine the volume of the regular tetrahedron ABCD . 2. Intersections of objects in space E.2770 Let ABCDEFGH be a cube. Con-sider a light source M placed above the cube such that : −−→ DH = −−→ HM Draw the shadow created by this light source around the cube. E.577 In space, consider the tetrahedron ABCD . We denote M , N , P points belonging to the edges [ DA ] , [ DC ] , [ DB ] respectively: Draw the intersection of the plane ( ABC ) and the plane ( MNP ) . 3. Using theorems E.4945 Consider the rectangular parallelepiped ABCDEFGH shown below : https://chingmath.fr chapExoCorrec/4943 sacados/4943 ABCD chapExoCorrec/2770 sacados/2770 MABCDEFGH chapExoCorrec/577 sacados/577 ABCDMNP chapExoCorrec/4945 sacados/4945
ABCDEFGHI SABCDEFM ABCD ABCDIJ ABCDEM IJMABCDE Point I is the midpoint of segment [ AE ] . The middles of the various edges are shown on the figure. Show the section of the ( DIF ) plane and the parallelepiped. Justify your construction. E.4944 Consider the pyramid ABCDEFS with vertex S whose base is a regular hexagon. Note M a point on segment [ SF ] . Draw the section plane of the pyramid with plane ( P ) parallel to its base plane passing through point M . E.582 In space, consider the regular tetrahedron ABCD : all its faces are equilateral triangles. 1 a Draw the height of triangle ABC originating from vertex C . b Draw the height of triangle ABD originating from ver-tex D . 2 Demonstrate that the straight lines ( AD ) and ( BC ) are orthogonal. E.589 In space, consider the tetrahedron ABCD ; I and J are the respective middles of the [ AD ] and [ BD ] edges. [ AD ] (resp. [ BD ] ) . 1 Show that the line ( IJ ) is parallel to the line ( AB ) . Trace the segment [ IJ ] . 2 Using similar reasoning, draw the point K midpoint of segment [ CD ] . 3 Show that the two planes ( IJK ) and ( ABC ) are parallel? E.4946 Consider the pyramid ABCDE with vertex E whose base is a trapezoid with ( BC ) == ( AD ) . Let M be a point on segment [ BE ] . Draw the section of the pyramid through the plane ( ADM ) . Justify your construction. 4. Problems E.578 In space, consider the pyramid ABCDE with a square base ; let I and J be the midpoints of segments [ BC ] and [ CE ] , respectively; M is a point on edge [ AB ] : https://chingmath.fr ABCDEFGHI chapExoCorrec/4944 sacados/4944 SABCDEFM chapExoCorrec/582 sacados/582 ABCD chapExoCorrec/589 sacados/589 ABCDIJ chapExoCorrec/4946 sacados/4946 ABCDEM chapExoCorrec/578 sacados/578 IJMABCDE
N-80-60-40-200204060-120-100-40-20020406080ABC N-80-60-40-200204060-120-100-40-20020406080AB ABCEFGKMNPUSVO−i−j−k 1 Show that ( EB ) is parallel to the plane ( IJM ) . 2 Use this to determine the intersection of planes ( ABE ) and ( IJM ) . Let N denote the point of intersection of plane ( IJM ) with line segment [ AE ] . 3 a Justify that lines ( AD ) and ( IM ) are intersecting. b Locate point T , the point of intersection of lines ( AD ) and ( MI ) . c Use this to determine the position of point P , the point of intersection of line ( DE ) and plane ( IJM ) . 4 Draw the cross-section of the plane through the pyramid. 5 Find point P in another way 5. Locating in the sphere E.6999 The meridians and parallels of the globe are shown below : Determine the geodesic coordinates of the points A , B and C . E.7000 Below is the globe with its geodetic ref- erence frame (meridians and parallels) : 1 Determine the geodesic coordinates of points A and B . 2 Taking 6 370 km for the radius of the earth, determine the distance as the crow flies separating points A and B rounded to the nearest kilometer. 6. Solid cross-sections E.6961 A private individual is inter-ested in the shadow cast on his future veranda by the roof of his house when the sun is at its zenith. This veranda is schematized below in cavalier perspective in an orthonormal reference frame O ; −→ i ; −→ j ; −→ j . The veranda roof consists of two triangular faces SEF and SFG . The ( SOA ) and ( SOC ) planes are perpendicular. ( SOC ) and ( EAB ) planes are parallel, as are ( SOA ) and ( GCB ) planes. [ UV ] and [ EF ] edges of roofs are parallel. https://chingmath.fr chapExoCorrec/6999 sacados/6999 N-80-60-40-200204060-120-100-40-20020406080ABC chapExoCorrec/7000 sacados/7000 N-80-60-40-200204060-120-100-40-20020406080AB chapExoCorrec/6961 sacados/6961 ABCEFGKMNPUSVO−i−j−k
ABCDEFGHMNP ABCDEFI Without calculation, justify that : 1 the segment [ KM ] is parallel to the segment [ UV ] ; 2 the segment [ NP ] is parallel to the segment [ UK ] . 7. Tracing cross-sections of solids E.6814 Consider a cube ABCDEFGH given below. Note M the middle of segment [ EH ] , N that of [ FC ] and P the point such that : −−→ HP = 1 4 · −−→ HG 1 Justify that the straight lines ( MP ) and ( FG ) intersect at a point L . Construct the point L . 2 We admit that the straight lines ( LN ) and ( CG ) are se-cant and we note T their point of intersection. We admit that the straight lines ( LN ) and ( BF ) are se-cant and we note Q their point of intersection. a Construct the points T and Q leaving the construction lines apparent. b Construct the intersection of planes ( MNP ) and ( ABF ) . 3 Deduce a construction of the cube’s section through the ( MNP ) plane. E.6819 Consider a solid ADECBF made up of two identical pyramids whose common base is the square ABCD of center I . A perspective representation of this solid is given below. All edges are of length 1 . We name M the middle of segment [ DF ] and N that of seg-ment [ AB ] . 1 Demonstrate that the planes ( FDC ) and ( ABE ) are par-allel. 2 Determine the intersection of planes ( EMN ) and ( FDC ) . 3 Construct the section of solid ADECBF by plane ( EMN ) . https://chingmath.fr chapExoCorrec/6814 sacados/6814 ABCDEFGHMNP chapExoCorrec/6819 sacados/6819 Extrait Liban 2016 ABCDEFI
ABCDMNP ABCDEFGHIJN E.2772 Consider the tetrahedron ABCD shown below : The points M , N , P belong to the planes ( ABC ) , ( ACD ) , ( ABD ) , respectively. Part A : Draw the intersection of ( MNP ) and ( BCD ) 1 a Draw the line ( d ) intersection of the planes ( AMP ) and ( BCD ) . Justify your construction. Name M and P the points of intersection of the straight line ( d ) with the straight lines ( BC ) and ( BD ) respectively. b Without justification, name X the point of intersection of the straight lines ( MP ) and ( d ) . 2 a Without justification, draw the line ( d ) intersection of the planes ( APN ) and ( BCD ) . Name N the point of intersection of the straight lines ( d ) and ( CD ) . b Justify that the straight lines ( NP ) and ( d ) are se-cant. Name Y the point of intersection of the straight lines ( NP ) and ( d ) . 3 Justify that the line ( XY ) is the line of intersection of the planes ( MNP ) and ( BCD ) . Draw the straight line ( XY ) . Part B: Draw the section of the tetrahedron. 4 a Place the point M  intersection of the line ( BC ) with the line ( XY ) . b Justify that the line ( MM  ) is the line of intersection of the planes ( ABC ) and ( MNP ) . Draw the straight line ( MM ) . c Place the point N  intersection of the line ( CD ) and the line ( XY ) . It is accepted that the line ( NN  ) is the line of intersection of the planes ( ACD ) and ( MNP ) . 5 Draw the section of the ( MNP ) plane with the tetrahe-dron. E.2769 Consider the cube ABCDEDFGH and the three points I , J , N belonging respectively to the edges [ EH ] , [ HG ] , [ AE ] ; we call ( P ) the plane ( IJN ) : 1 a Draw the plane ( P ) passing through J and parallel to the plane ( EHD ) . b Draw the line (Δ) of intersection of the planes ( P ) and ( P ) . c Place the point N intersection of the line (Δ) with the plane ( EFB ) . d Deduce the position of the point M , intersection of the plane ( P ) with the straight line ( AB ) . 2 Place the point L , intersection of the line ( BC ) with the plane ( P ) . 3 Place the point K , intersection of the line ( CG ) with the plane ( P ) . 4 Draw the section of the plane P with the cube. https://chingmath.fr chapExoCorrec/2772 sacados/2772 ABCDMNP chapExoCorrec/2769 sacados/2769 ABCDEFGHIJN
ABCDEKJI ABCDEFGHIJ r495-1 KABCDEFGHIJ E.2765 Consider the ABCDE square-based pyramid shown below. The points I , J , K are the respec-tive middles of the edges [ CE ] , [ BC ] , [ AB ] : 1 a Justify that the straight lines ( BE ) and ( IJ ) are parallel. b Specify the position of the line ( d ) of intersection of the planes of the ( ABE ) and ( IJK ) planes. Then plot the line ( d ) . Note L the point of intersection of the straight lines ( d ) and ( AE ) . Note that the point L belongs to the plane ( IJK ) . 2 In this question, we will study the intersection of the plane ( IJK ) with the edge [ ED ] : a Determine the location of point T , intersection of plane ( IJK ) with line ( AD ) . b Justify that the straight line ( LT ) belongs to the plane ( ADE ) . c Deduce the position of point M , intersection of plane ( IJK ) with edge [ ED ] . 3 Represent the section of the pyramid ABCDE with the plane ( IJK ) . E.2794 Consider the cube ABCDEFGH below and the points I , J respective middles of the edges [ AD ] and [ DC ] : 1 a Determine the position on the figure of the point M intersection of the line ( AB ) with the plane ( IJF ) . b Determine the position on the figure of the point P intersection of the line ( AE ) with the plane ( IJF ) . 2 a Determine point N , intersection of line ( BC ) with plane ( IJF ) . b Determine the point Q , intersection of the straight line ( GC ) with the plane ( IJF ) . 3 Draw on the figure above the section of the cube with the ( FIJ ) plane. Voir en animation la correction: 8. Annales E.6902 Consider the cube ABCDEFGH shown below. We define the points I and J respectively by: −→ HI = 3 4 · −−→ HG ; −→ JG = 1 4 · −−→ CG 1 On the figure below, draw, without justification, the sec-tion of the cube through the plane ( IJK ) where K is a point on the segment [ BF ] . 2 On the figure below, draw, without justification, the sec- https://chingmath.fr chapExoCorrec/2765 sacados/2765 ABCDEKJI chapExoCorrec/2794 sacados/2794 ABCDEFGHIJ r495-1 chapExoCorrec/6902 sacados/6902 KABCDEFGHIJ
LABCDEFGHIJ ABCDEOS tion of the cube through the plane ( IJL ) where L is a point on the line ( BF ) . 3 Is there a point P on the line ( BF ) such that the section of the cube through the plane ( IJP ) is an equilateral triangle? Justify your answer. 9. Unclassified financial years E.3123 In space provided with the O ; −→ i ; −→ j ; −→ k orthonormal direct, we consider points A of coordinates 0 ; 0 ; 8 , B of coordinates 0 ; 0 ; 8 , C of coor-dinates 4 ; 0 ; 8 . 1 a Make the figure with the points defined in exercise (graphic unit : 1 cm ) . b Demonstrate that : The straight lines ( BC ) and ( BA ) are orthogonal ; The straight lines ( CO ) and ( OA ) are orthogonal ; The line ( BC ) is orthogonal to the plane ( OAB ) . c Determine the volume, in cm 3 , of the tetrahedron OABC . d Show that the four points O , A , B , C lie on a sphere whose center and radius will be determined. 2 To any real k of the open interval 0 ; 8 , is associated the point M 0 ; 0 ; k . The plane ı which contains M and is orthogonal to the straight line ( OB ) meets the straight lines ( OC ) , ( AC ) , ( AB ) at N , P , Q respectively. a Determine the nature of the quadrilateral ( MNPQ ) . is b line ( PM ) orthogonal to line ( OB ) ? For what value of k , is the line ( MP ) orthogonal to the line ( AC ) ? c Determine MP 2 as a function of k . For what value of k , is the distance PM minimum? E.6879 In space, consider a pyramid SABCE with square base ABCE center O . Let D be the point in space such that O ; −→ OA ; −−→ OB ; −−→ OD is an orthonor-mal datum. The point S has coordinates 0 ; 0 ; 3 in this frame. 1 Let U be the point on the line ( SB ) of dimension 1 . Con-struct the point U on the figure. 2 Let V be the point of intersection of the plane ( AEU ) and the line ( SC ) . Show that the straight lines ( UV ) and ( BC ) are parallel. Construct the point V on the figure. 3 Assume that the point K 5 6 ; − 1 6 ; 0 is the foot of the height from U in the trapezoid AUV E . Determine the area of the trapezoid AUV E . https://chingmath.fr LABCDEFGHIJ sacados/3123 chapExoCorrec/6879 sacados/6879 ABCDEOS