Outside the high school program / Cartesian geometry 42 exercises (including 33 corrected)

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IJO xxyy-6-5-4-3-2-123I-2-12345678JO 1. Right-angled triangle and median E.943 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the following two points : A ( − 4 ; − 2) ; B ( − 1 ; 2) 1 Place points A and B . The graph will be completed as the questions the exercise. 2 Let K be the midpoint of segment [ AB ] . Show that the point K has coordinates : K ( − 2.5 ; 0) . 3 Consider the point C with coordinates ( − 2.5 ; − 2.5) . a Determine lengths AB and KC . b What does the segment [ KC ] represent for the triangle ABC ? c Deduce that the triangle ABC is right-angled at C . E.947 The plane is provided with an orthonormal reference frame ( O ; I ; J ) of unit 1 cm , place the following points : A ( − 4 ; − 2) ; B ( − 1 ; − 5) ; C (4 ; 0) 1 Construct the reference frame O ; I ; J and place the points A , B , C . 2 Show that : AC =2 17 . 3 Let M be the midpoint of segment [ AC ] . Determine the coordinates of point M . 4 Determine the length of segment [ BM ] . 5 Deduce the nature of the triangle ABC . E.2742 In the plane provided with a ( O ; I ; J ) orthonormal, consider the circle C of center K (2 ; 1) and ra-dius 2.5 , and the point A 0 ; 5 2 1 Show that the point A belongs to the circle C . 2 Determine the coordinates of the point B diametrically opposite the point A in the circle C . 3 Let C 3 2 ; 1 − 6 , justify that the triangle ABC is right-angled at C . 4 Determine the coordinates of a point on the circle C , whose abscissa is 5 2 E.4595 Consider the plane provided with a O ; I ; J orthonormal coordinate system. Consider the three points : A ( − 1 ; − 2) ; B (3 ; 4) ; C 2 ; 1 − 2 3 1 Demonstrate that the triangle ABC is rectangular at C . 2 Determine the coordinates of point D midpoint of seg-ment [ AB ] . 3 Consider the point E with coordinates ( 1 ; 1 −√ 13 ) . a Determine the measure of segment [ DE ] . b Demonstrate that the triangle ABE is right-angled. 4 a Determine the coordinates of the point F diametri-cally opposite C in the circle of diameter [ AB ] . b Show that the quadrilateral AFBC is a rectangle. E.6511 Consider the plane provided with an or-thonormal reference frame O ; I ; J . The points A , B and C have coordinates : A ( − 1 ; − 1) ; B (2 ; 1) ; C ( − 2 ; 7) 1 Justify that the point I midpoint of the segment [ AC ] has coordinates I − 3 2 ; 3 . 2 a Determine the coordinates of the point D so that the point I is the midpoint of the segment [ BD ] . b Represent the quadrilateral ABCD in the reference frame below. c Justify that the quadrilateral ABCD is a parallelo-gram. 3 We admit that AC = 65 . Justify that the quadrilateral ABCD is a rectangle. https://chingmath.fr chapExoCorrec/943 sacados/943 IJO chapExoCorrec/947 sacados/947 Groupe Est - Septembre 2003 - 6 points chapExoCorrec/2742 sacados/2742 chapExoCorrec/4595 sacados/4595 chapExoCorrec/6511 sacados/6511 xxyy-6-5-4-3-2-123I-2-12345678JO
ABCDEFG ABCDIJ ABCDMNO ABIJMNOC OABCDMNC 2. isometric triangles E.553 The figure opposite is made up of the tri-angle ABC on which we have constructed two squares outside this triangle: EABD and ACFG . 1 Show that the triangles EAC and GAB are isometric triangles. 2 Deduce that : BG = EC . E.568 Let ABCD be a parallelogram. Let I denote the midpoint of [ AB ] and J denote the midpoint of [ DC ] . 1 Show that triangles ADJ and CBI are isometric. 2 Conclude that : AJ = CI . E.571 Let ABCD be a parallelogram of center O and M a point on the segment [ AB ] . Note N the point of intersection of ( OM ) and ( DC ) . 1 Without isometry: a Show that the triangles MBO and NDO are isometric triangles b Deduce that O is the midpoint of segment [ MN ] . c What can you say about the quadrilateral BNDM . 2 With isometries : a Show that point O is the midpoint of segment [ MN ] . b Show that the triangles AMO and CNO are isometric. E.567 Let C be a circle with center O , and A , B two points of the circle. On the chord [ AB ] of the circle, we place the points I and J such that : AI = IJ = JB Note respectively M and N the points of intersection of the half-lines [ OI ) and [ OJ ) with the circle C . 1 What is the nature of the triangle ABO ? 2 Show that the triangles AIO and OJB are isometric. 3 What is the nature of the triangle IJO ? 4 Show that the straight lines ( IJ ) and ( MN ) are parallel E.556 Let C be a circle of center O . Consider the square ABCD having its vertices A and C on the circle. We’ll use the properties of inscribed angles and angles at the center. 1 Show that [ MN ] is a diameter of C . 2 Show that : NOC =90 o What can we say about the lengths CM and CN ? 3 Noting that NCM =90 o , compare the two angles MCB and NCD . 4 Deduce that the triangles NDC and MCB are isometric. We’ve just established that the following equality of lengths : ND = BM https://chingmath.fr chapExoCorrec/553 sacados/553 ABCDEFG chapExoCorrec/568 sacados/568 ABCDIJ chapExoCorrec/571 sacados/571 ABCDMNO chapExoCorrec/567 sacados/567 ABIJMNOC chapExoCorrec/556 sacados/556 OABCDMNC
ABCDEFO ABCDE ABCMNP CDBAMC E.569 Consider the following configuration : 1 Draw a figure in your notebook that has the same prop-erties as the one above. ABCD est un parallélogramme O est le centre de ce parallélogramme E est la projection orthogonale de D sur la droite ( AC ) F est la projection orthogonale de B sur la droite ( AC ) 2 What can be said about angles EOD and BOF ? 3 Show that triangles ODE and OBF are isometric. 4 Conclude that : OE = OF . E.555 Let ABC be an isosceles triangle in A and Δ be the perpendicular bisector of segment [ AB ] . Note D the point of intersection of the lines Δ and ( BC ) . We place the point E on the straight line ( AD ) as shown opposite and verifying AE = CD . 1 a What can be said about the angles DAB , DBA and ACB ? Justify. b Compare the angles DCA and EAB . 2 Deduce that the triangles ADC and AEB are isometric E.559 Let ABC be an equilateral triangle. Let M , N and P be three points belonging re-spectively to the segments [ AB ] , [ BC ] and [ AC ] such that : AM = BN = CP 1 Demonstrate that the triangles APM , BMN and CNP are isometric triangles 2 Deduce that the triangle MNP is an equilateral triangle. 3. similar triangles E.566 In the figure opposite, the points A , B , C and D are four points on the circle C . Show that the triangles DCM and ABM are similar. E.570 Let three points E , D and F be three points on a circle C . The bisector of the angle EDF inter-sects C at K and intersects the straight line ( EF ) at M . 1 Justify the equality of the angles EDK and EFK . 2 Justify the equality of angles DEF and DKF . 3 Deduce that the triangles DEM and FKM are similar. https://chingmath.fr chapExoCorrec/569 sacados/569 ABCDEFO chapExoCorrec/555 sacados/555 ABCDE chapExoCorrec/559 sacados/559 ABCMNP chapExoCorrec/566 sacados/566 CDBAMC chapExoCorrec/570 sacados/570
ABCMN OABCDE ABCEF IJKABCMPQ¸o¸o E.560 Let ABC be an isosceles triangle at A . 1 Note I and J the respective middles of segments [ AC ] and [ AB ] . Show that : BI = CI 2 The bisector of the angle ABC intersects [ AC ] at K and the bisector of the angle ACB intersects [ AB ] at L . Show that : BK = CL 3 The height from B , cuts [ AC ] into M and the height from C , cuts [ AB ] into N . Show that : BM = CN E.563 Let ABC and AMN be two tri-angles such that the straight lines ( BC ) and ( MN ) are parallel and such that : AB = 9 4 ; BC = AM = 3 2 1 Calculate the length of seg-ment [ MN ] . 2 Draw the height from A of the triangle ABC . Note H the foot of the height. Note H the foot of the height of triangle AMN from A . Jus-tify. 2 With your ruler, measure the length AH . Then, using Thales’ theorem, determine the length AH . 3 Calculer le rapport A ire ( ABC ) A ire ( MNP ) . Demonstrate that this ratio is equal to 9 4 . E.554 Let C be a circle with center O and A a point outside C . Draw two rays through A , each of which intersects the circle at two points. The graph below : 1 Show that : ACD = BEA 2 What can you say about triangles ACD and ABE ? Jus-tify your answer. 3 a Show that : AD × BE = AB × DC b Suppose that : AD = 3 ; 7 cm ; AB = 3 ; 3 cm ; DC = 6 ; 8 cm Use this to determine the measure of BE to the nearest millimeter. E.561 Let ABC be a triangle. We construct out-side the triangle ABC two right-angled and isosceles triangles in A . 1 Show that : FB = CE . 2 Note I and J the respective middles of the segments [ FB ] and [ CE ] . Show that A lies on the perpendicular bisector of segment [ IJ ] E.4723 In a triangle ABC , consider a point M , interior to this triangle, realizing the following equality: CAM = CBM Let P and Q be the orthogonal projects of the point M onto the line ( AC ) and the line ( BC ) respectively. Note I , J and K the respective middles of the segments [ AB ] , [ AM ] and [ BM ] . 1 a Justify the equality: AJ = JP . b Establish equality: JP = IK . 2 Establish equality: IJ = KQ . 3 a Justify that the quadrilateral IJMK is a parallelo-gram. b Show that the angles PJI and IKQ are of equal mea-sures. 4 Deduce that the triangle IPQ is an isosceles triangle. https://chingmath.fr sacados/560 sacados/563 ABCMN chapExoCorrec/554 sacados/554 OABCDE chapExoCorrec/561 sacados/561 ABCEF sacados/4723 IJKABCMPQ¸o¸o
ABCDIxMNPQ OABIJH E.4901 Consider a square ABCD of direct direction. The point M is a point on the half-line [ BC ) not belonging to the segment [ BC ] . Note N the point of intersection of the straight lines ( CD ) and ( AM ) , and P the point of intersection of the line ( BC ) and the line passing through A perpendicular to ( AM ) . We define the point Q as the midpoint of the segment [ PN ] . We note I the center of square ABCD . 1 a Show that the angles DAN and BAP are of equal measures. b Show that the triangles ADN and ABP are isometric. 2 a Show the following equality: DN = DC . b Demonstrate the triangles ABQ and CBQ are isomet-ric. 3 Deduce the geometric locus of the point Q when the point M describes the half-line [ Cx ) . 4. Sangaku E.6398 Here is a representation of a san-gaku: It’s made up of two half-discs and a disc all tangent to each other. Determine the measures IH and OH as a function of the length JI . 5. Geometric location E.4650 In the plane, consider a circle C with center O and [ AB ] a diameter of this circle. The point C is a point of the circle C distinct from A and from B . To any point M of the circle C , distinct from A and B , we associate the following construction : the line ( d ) parallel to the line ( AC ) passing through the point M ; it intercepts the circle C at D a second time. the straight line ( d ) parallel to the straight line ( BC ) passing through the point M ; it intercepts the circle C a second time at E . we note G the center of gravity of the triangle MDE . Specify the geometric locus of the point G when the point M moves on the circle C . E.6032 Consider a square whose two hypotenuse vertices lie on the x-axis and y-axis respectively. When the square slides under these conditions, what is the geometric locus of the vertex of the right angle? 6. Circles and tangents https://chingmath.fr sacados/4901 A verifier ABCDIxMNPQ chapExoCorrec/6398 sacados/6398 Sangaku OABIJH chapExoCorrec/4650 sacados/4650 sacados/6032
OCABCP ABCMNP OIMN(d1(d2(DCA E.1450 Consider the configuration given below : 1 Using the square, check that the straight line (Δ) is a tangent of the circle C with center O . 2 Draw the circle C with center P and tangent to the line (Δ) . Through quel (s) point (s) passe (ent) of the figure, does the circle C pass...? E.1094 Let C be a circle with center O and A a point outside the circle C . Note C the circle with diameter [ OA ] . Note M and N the two points of intersection of the circles C and C . 1 Make a figure representing this configuration. 2 What can be said about the straight line ( AM ) relative to the circle C ? Justify your assertion. E.1093 Consider the following configuration : ˇ Let ( d ) be a straight line and H a point on this straight line. C is a circle tangent to the line ( d ) having as its point of contact the point H .ı Draw such a configuration and indicate a construction method. E.2936 In the plane, consider the triangle ABC right-angled B and M a point on the segment [ AC ] such that AMB is a right angle; the points N and P are the symme-tries of the point M , respectively with respect to the straight lines ( BC ) and ( AB ) : 1 a Justify the following length equalities: BM = BN = BP b Show that : PBN =180 o . c Justify that the circle C of diameter [ NP ] admits the straight line ( AC ) as tangent at the point M . 2 a Demonstrate that the points B , C , M , N are co-cyclic to a circle we’ll call C . b Give the position of the line ( AB ) relative to the circle C . E.1840 Consider a circle C , a point O and the two straight lines ( d 1 ) and ( d 2 ) tangent to the circle passing through the point O . Consider a straight line ( D ) passing through O and lying be-tween the straight lines ( d 1 ) and ( d 2 ) : we are free to place the line ( D ) at any point, but subject to these two constraints. The aim of this exercise is to determine the set described by I when the straight line ( D ) describes the set of straight lines passing through O and lying between ( d 1 ) and ( d 2 ) : 1 a Where is the point I when the straight line ( D ) is such that the points M and N are diametrically op-posed? b Draw the line ( D ) at three different points and the associated point I . 2 a Make a conjecture as to the set of points described by the point I . b Establish this conjecture. 7. Trigonometry reminder https://chingmath.fr chapExoCorrec/1450 sacados/1450 OCABCP chapExoCorrec/1094 sacados/1094 chapExoCorrec/1093 sacados/1093 chapExoCorrec/2936 sacados/2936 ABCMNP chapExoCorrec/1840 sacados/1840 OIMN(d1(d2(DCA
ABCx ¸˛ACB IJO E.531 Consider the right-isosceles triangle at C opposite. Let x be the measure of side AC . 1 Complete the table : ACB CAB Measurement in degrees 2 a Using Pythagoras’ theorem, express the measure of side [ AB ] as a function of x . b In the right-angled triangle ABC , determine the sine, cosine and tangent of the angle CAB . c Complete the table : ¸ cos ¸ sin ¸ tan ¸ 45 o E.537 Consider a triangle ABC right-angled C . We note : ¸ = CAB ; ˛ = ABC 1 According to the measures of the sides of the triangle ABC : a Express the values of cos ¸ and sin ˛ . b Compare the values of tan ¸ and tan ˛ . 2 Justify that for ¸ ∈ 0 ; 90 , we have : cos ¸ = sin ı 2 − ¸ tan ı 2 − ¸ = 1 tan ¸ 3 By expressing cos ¸ and sin ¸ in terms of the measures of the sides of the triangle ABC and using the Pythagorean theorem, demonstrate the following formula : cos ¸ 2 + sin ¸ 2 = 1 8. median theorem E.8103 Reminder : in the plane equipped with an orthonormal basis O ; I ; J , let A ( x A ; y A ) and B ( x B ; y B ) be : The distance AB is defined by: AB = x B − x A 2 + y B − y A 2 Let I be the midpoint of the segment [ AB ] . The coor-dinates of the point I are: I x A + x B 2 ; y A + y B 2 In the plane equipped with an orthonormal coordinate system O ; I ; J , consider the following two points : A ( − 4 ; − 2) ; B ( − 1 ; 2) 1 Place the points A and B . The graph will be completed as the questions in the exercise are answered. 2 Let K be the midpoint of the segment [ AB ] . Show that the coordinates of the point K are: K ( − 2 ; 5 ; 0) . 3 Consider the point C with coordinates ( − 2 ; 5 ; − 2 ; 5) . a Determine the lengths AB and KC . b What does segment [ KC ] represent for triangle ABC ? c Deduce that triangle ABC is a right triangle at C . https://chingmath.fr chapExoCorrec/531 sacados/531 ABCx chapExoCorrec/537 sacados/537 ¸˛ACB chapExoCorrec/8103 sacados/8103 IJO
-6-5-4-3-2-1234I-3-2-1234JO 120m77m8mdEntréeSortie ABCDMABCDMABCDM xyCABMNP4cm E.5292 Consider the plane provided with an orthonormal reference frame ( O ; I ; J ) : 1 Place the three points A , B , C in the marker below : A (3 ; − 3) ; B ( − 4 ; 3) ; C ( − 5 ; − 1) 2 Determine the coordinates of the middle M of segment [ AB ] . 3 a Determine the lengths AB and MC . b Establish that the triangle ABC is right-angled at C . 4 Let N be a point on the ordinate axis. Determine the coordinates of the point N so that the vectors −−→ BN and −−→ CM are collinear. E.3038 Consider the triangle ABC equilateral whose sides measure 6 cm ; note I , J , K the respective mid-dles of [ BC ] , [ AC ] , [ AB ] ; M is the center of gravity of the triangle ABC . 1 Determine the length of segment [ BJ ] and [ BM ] . 2 Determine the value of the following scalar products : a −→ AC · −−→ AB b −→ AC · −→ IC c −−→ MC · −−→ MA d −−→ CM · −−→ MI 9. Unclassified financial years E.6031 The city ˇ Promenade ı wishes to move into a square-shaped plot of land with a river running through it laterally. The figure below repre-sents the plot and the river is the hatched part : At what distance d should the bridge be placed so that the distance traveled by a visitor is minimal? E.6030 Consider a square ABCD and a point M belonging to segment [ AB ] . Inside the square we construct : a square with segment [ AM ] as side ; an isosceles triangle admitting segment [ MB ] for main base. Consider the domain D of the plane formed by these two fig-ures. Here are three representations of this situation : Determine the position of the point M for the area of the domain D to be half that of the square ABCD . E.6033 The figure opposite shows the semi-circle C with diameter [ AB ] , where the axes ( Ax ) and ( By ) are perpen-dicular to the line ( AB ) . M is any point on the half-line [ Ax ) . N is the point of intersection, other than B , of the semicircle C and the line ( MB ) . P is the point of intersection of the line ( AN ) and the axis ( By ) . What can be said about the function f defined by: f : AM −→ BP E.6034 Two balls lie at the bottom of a cu-bic box with edge 15 cm . The large ball has a radius 3 times greater than the small ball. The cross-section given opposite shows that they are ˇ perfectly nested ı at the bottom of this box. Determine the radius of each of these balls. https://chingmath.fr chapExoCorrec/5292 sacados/5292 -6-5-4-3-2-1234I-3-2-1234JO chapExoCorrec/3038 sacados/3038 sacados/6031 120m77m8mdEntréeSortie sacados/6030 ABCDMABCDMABCDM sacados/6033 xyCABMNP4cm sacados/6034
OIJABCD -4-3-2-1234I-2-12JO -4-3-2-1234K-4-224LP -10-9-8-7-6-5-4-3-2-1M2345678910-5-4-3-2-1QN2345 E.4616 The plane is provided with a reference frame O ; I ; J of any kind shown below. Consider the four points A , B , C and D : 1 Donner les coordonnées des points A , B , C et D dans le repère O ; I ; J . 2 Demonstrate that the quadrilateral ABCD is a parallel-ogram. E.921 The plane is provided with an or-thonormal reference ( O ; I ; J ) . The unit of length is the cen-timeter. 1 a Place the point A (5 ; 3) . b Determine distance IA . 2 Consider the circle C of center I and radius 5 and the point B − 1 ; 21 . a Show that the points A and B belong to the circle C . b Draw the circle C and place the point B . 3 a Place the point C , symmetrical to A with respect to I . b Establish, without any calculation, that the triangle ABC is right-angled at B . 4 a Place the point D such that the quadrilateral ABCD is a rectangle b Determine by calculation the coordinates of the point D . E.4526 Consider the three landmarks below : O ; I ; J : P ; K ; L : Q ; M ; N : 1 Name each of these landmarks. 2 Consider the points A , B and C with coordinates : A (3 ; − 1) ; B (0 ; − 2) ; C (2 ; 2) a Place points A , B and C in each of the markers. b Check, using the square, that the triangle ABC is right-angled at A in the reference frame O ; I ; J . c What is the nature of the triangle ABC in the other two landmarks? https://chingmath.fr chapExoCorrec/4616 sacados/4616 OIJABCD chapExoCorrec/921 sacados/921 chapExoCorrec/4526 sacados/4526 -4-3-2-1234I-2-12JO -4-3-2-1234K-4-224LP -10-9-8-7-6-5-4-3-2-1M2345678910-5-4-3-2-1QN2345
-4-3-2-1234I-2-123JOABC E.8292 Consider the plane with a reference frame O ; I ; J and the three points A , B , C shown below : 1 Give the coordinates of the points A , B , C . 2 a Place the point D so that the quadrilateral ABCD is a parallelogram. b Give the coordinates of the point D . 3 a Place the point E so that the quadrilateral ABEC is a parallelogram. b Give the coordinates of the point E . https://chingmath.fr chapExoCorrec/8292 sacados/8292 -4-3-2-1234I-2-123JOABC