Outside the high school program / Cartesian geometry 42 exercises corrected

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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 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
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 chapExoCorrec/568 sacados/568 ABCDIJ chapExoCorrec/571 sacados/571 chapExoCorrec/567 sacados/567 chapExoCorrec/556 sacados/556
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 chapExoCorrec/559 sacados/559 chapExoCorrec/566 sacados/566 chapExoCorrec/570 sacados/570
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.4681 Consider a triangle ABC right-angled at B and such that : ABC = 34 o Note H the foot of the height, in triangle ABC , from vertex C . 1 Determine the measure of the angle CAB . 2 Demonstrate the following equality: CBH = ACH https://chingmath.fr sacados/560 sacados/563 chapExoCorrec/554 sacados/554 OABCDE chapExoCorrec/561 sacados/561 sacados/4681 vient de quatrieme ... a adapter pour la seconde ABCH34o
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. 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 . https://chingmath.fr sacados/4723 IJKABCMPQ¸o¸o sacados/4901 A verifier ABCDIxMNPQ chapExoCorrec/6398 sacados/6398 Sangaku OABIJH chapExoCorrec/4650 sacados/4650
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 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 . https://chingmath.fr sacados/6032 chapExoCorrec/1450 sacados/1450 chapExoCorrec/1094 sacados/1094 chapExoCorrec/1093 sacados/1093 chapExoCorrec/2936 sacados/2936
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 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) https://chingmath.fr chapExoCorrec/1840 sacados/1840 chapExoCorrec/531 sacados/531 ABCx chapExoCorrec/537 sacados/537 ¸˛ACB chapExoCorrec/8103 sacados/8103
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 . 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? https://chingmath.fr IJO chapExoCorrec/5292 sacados/5292 -6-5-4-3-2-1234I-3-2-1234JO chapExoCorrec/3038 sacados/3038 sacados/6031 120m77m8mdEntréeSortie
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. E.2958 In the plane provided with a reference frame ( O ; I ; J ) , consider the representative curve C f of the square function. 1 Let M be a point on the x-axis with abscissa x ; N is the point with abscissa x belonging to the perpendicular to the line ( JM ) passing through the point O . The aim of this question is to show that the point N belongs to the curve C f : a Express MQ as a function of ¸ and x . b Justify that the angle JOQ has measure ¸ . c Express QJ as a function of ¸ . d Justify the equality: tan ¸ = 1 x e Show that : MN = x 2 . f Justify that the point N belongs to the curve C f 2 Let A be a point on the curve C f of abscissa x ; note B the orthogonal project of the point A onto the x-axis. a Justify that : OJB = AOB b Deduce that BCO is a right angle. 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. https://chingmath.fr sacados/6030 ABCDMABCDMABCDM sacados/6033 xyCABMNP4cm sacados/6034 sacados/2958 A verifier chapExoCorrec/4616 sacados/4616 OIJABCD
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? 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/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 chapExoCorrec/8292 sacados/8292 -4-3-2-1234I-2-123JOABC