Grade 10 / Midpoint of a segment and norm of a vector 66 exercises (100% corrected)

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-22468101214I-4-224JOCD -5-4-3-2-12345I-3-2-12JO -4-3-2-12I-2-1JOABCD ChingQuizz : 7 exercises available for Quizz assessment : 1. Around the length E.941 Consider the plane provided with the reference frame ( O ; I ; J ) orthonormal below : 1 The purpose of this question is to determine the length of segment [ CD ] : a Give the coordinates of points C and D . b Place the point E (14 ; − 2) . What is the nature of the triangle CDE ? c Give the measures of segments [ CE ] and [ ED ] . d Using the Pythagorean theorem, determine the length of segment [ CD ] . 2 Place the points F ( − 2 ; 4) and G (13 ; − 4) in the refer-ence frame. Using a similar procedure, show that : FG =17 3 Let A and B be any two points in the plane with coordi-nates ( x A ; y A ) and ( x B ; y B ) respectively. Justify that the distance AB as a function of x A , x B , y A and y B is expressed as : AB = x B − x A 2 + y B − y A 2 4 Use the formula to establish that : CG = 125 E.8294 Consider the plane provided with the reference frame O ; I ; J shown below : 1 a Place the points A and B with coordinates : A ( − 3.5 ; − 2.5) ; B (1.5 ; − 2.5) b Give, graphically, the length of segment [ AB ] . 2 a Place the points D and C with coordinates : C (3.5 ; − 3) ; D (3.5 ; 1) b Give, graphically, the length of segment [ CD ] . 3 a Place the points E and F with coordinates : E ( − 4 ; − 2) ; F (1 ; 2) b Give the exact measurement of segment [ EF ] . All traces of research or initiative will be taken into account in the evaluation of the exercise 2. Vector standards E.8295 In a plane with an orthonormal coor-dinate system O ; I ; J , consider the six points shown below : 1 Give the measurement, in centimeters, of the unit of the marker. 2 a The segment [ AB ] has been subdivided into 5 equal parts. Check that segment [ AB ] measures 5 units. b By graphical reading, complete the blanks below : x A = : : : ; y A = : : : ; x B = : : : ; y B = : : : c Give the value of the expression below : x B − x A 2 + y B − y A 2 Proposal: consider the plane provided with an orthonor-mal reference frame and A ( x A ; y A ) and B ( x B ; y B ) . The distance AB has the value : AB = x B − x A 2 + y B − y A 2 2 a Complete the blanks to obtain the expression giving the measure of the segment [ CD ] : https://chingmath.fr chapExoCorrec/941 sacados/941 -22468101214I-4-224JOCD chapExoCorrec/8294 sacados/8294 -5-4-3-2-12345I-3-2-12JO chapExoCorrec/8295 sacados/8295 -4-3-2-12I-2-1JOABCD
-6-5-4-3-2-123456I-2-123JO -123I-3-2-1234JO CD = : : : − : : : 2 + : : : − : : : 2 b Give the measure of segment [ CD ] . E.11473 Reminder : In a plane equipped with an orthonormal co-ordinate system, consider the two points A ( x A ; y A ) and B ( x B ; y B ) . The vector −−→ AB has the following coordinates : −−→ AB ( x B − x A ; y B − y A ) Proposition: In a plane equipped with an orthonormal coordinate system, consider a vector −→ u with coordinates −→ u ( x ; y ) . The norm of the vector −→ u has the following value : −→ u = x 2 + y 2 In the plane equipped with an orthonormal coordinate sys-tem, consider the three points : A (2 ; − 10) ; B (20 ; 14) ; C (23 ; 10) 1 Determine the coordinates of the vectors −−→ AB , −→ AC , and −−→ BC . 2 Establish the norms : −−→ AB = 30 ; −→ AC = 29 ; −−→ BC = 25 E.11474 In a plane equipped with an or-thonormal coordinate system, consider the following three points : A ( − 3 ; − 4) ; B (2 ; 8) ; C (12 ; 32) 1 Determine the coordinates of vectors −−→ AB , −→ AC , and −−→ BC . 2 Establish the norms : −−→ AB = 13 ; −→ AC = 39 ; −−→ BC = 26 E.9601 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the two points E ( − 1 ; − 2) ; F (2.5 ; − 1.5) . Determine length EF . E.8296 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the two points A and B whose coordinates are: A 11 3 ; 3 ; B 5 3 ; 3 2 Determine the measure of segment [ AB ] . 3. Length and isosceles triangle E.4523 In the plane equipped with the orthonormal reference frame O ; I ; J , consider the three points A , B , and C with coor-dinates : A ( − 3 ; − 2) ; B (2 ; 0) ; C ( − 1 ; 3) 1 a Determine the coordinates of vector −−→ AB . b Deduce that : −−→ AB = 29 2 Determine the norm −→ AC . 3 a Place the points A , B , and C in the coordinate sys-tem above : b Justify that the triangle ABC is isosceles at A . E.4525 Consider the plane equipped with an orthonormal coordinate system O ; I ; J and the three points : A (3 ; 1) ; B (1 ; 2) ; C ( − 1 ; − 2) 1 Place the points A , B , and C in the coordinate system above. 2 a Determine the norms of the vectors −−→ AB , −→ AC , and −−→ BC . b Establish that triangle ABC is a right triangle. Specify the vertex of its right angle. E.4524 Equip the plane with an orthonor-mal basis O ; I ; J . Consider the three points : A (1 ; 2) ; B (2 ; − 1) ; C ( − 2 ; 1) 1 Determine the norm of vectors −−→ AB and −→ AC . 2 Establish that triangle ABC is isosceles at A . E.2706 In the plane equipped with an or-thonormal reference frame ( O ; I ; J ) , we consider the three points A , B , C with respective coordinates : A ( − 1 ; − 1) ; B (2 ; 3) ; C 9 2 ; − 2 . 1 Determine the norms of vectors −→ AC and −−→ BC . 2 Establish that triangle ABC is isosceles at C . 4. Length and right triangle https://chingmath.fr chapExoCorrec/11473 sacados/11473 chapExoCorrec/11474 sacados/11474 chapExoCorrec/9601 sacados/9601 chapExoCorrec/8296 sacados/8296 chapExoCorrec/4523 sacados/4523 -6-5-4-3-2-123456I-2-123JO chapExoCorrec/4525 sacados/4525 -123I-3-2-1234JO chapExoCorrec/4524 sacados/4524 chapExoCorrec/2706 sacados/2706
-2246I-6-4-22JO IJOCfFMP E.11531 In an orthonormal coordinate system O ; I ; J , consider the three points : A ( − 3 ; 5) ; B (28 ; 5 ; 13) ; C (21 ; 23) 1 Establish the following equalities: −−→ AB = 32 ; 5 ; −→ AC = 30 2 a Determine the length of the segment [ BC ] . b Deduce that triangle ABC is a right triangle. E.11534 In an orthonormal coordinate system O ; I ; J , consider the following three points : A ( − 3 ; 5) ; B (25 ; 21 ; 5) ; C (7 ; 29) 1 Establish the following equalities: −−→ AB = 32 ; 5 ; −→ AC = 26 2 a Determine the length of the segment [ BC ] . b Deduce that triangle ABC is a right triangle. E.9600 Equip the plane with an orthonormal coordinate system O ; I ; J . Consider the following three points : D ( − 3 ; − 1) ; E ( − 2 ; − 2) ; F (0 ; 2) Prove that triangle DEF is a right triangle at D . 5. Length and quadrilateral E.2705 Consider the plane equipped with an or-thonormal coordinate system ( O ; I ; J ) and the four points A , B , C , D with respective coordinates : A ( − 2 ; − 3) ; B (0 ; 1) C (6 ; − 2) ; D (4 ; − 6) . 1 Place these four points on the coordinate system below : 2 a Determine the exact lengths of the four sides of the quadrilateral ABCD . b Establish that the quadrilateral ABCD is a parallelo-gram. 3 Demonstrate that ABCD is a rectangle. E.10623 In the plane equipped with an or-thonormal coordinate system, consider the four points : A ( − 10 ; − 5) ; B ( − 8 ; 3) ; C (14 ; 2) ; D (12 ; 18) 1 Determine −→ AC and −−→ BD . 2 Establish that the quadrilateral ABCD is not a rectan-gle. 6. Length and geometry E.8039 Consider the plane with a reference frame O ; I ; J and the circle C with center A (3 ; 2) and ra-dius 5 . 1 Which of the points B (6 ; 6) and C (2 ; 7) belong to the circle C : 2 Represent this configuration to check your answers. E.8108 In the plane provided with a reference frame O ; I ; J , consider the circle C of center A ( − 3 ; − 2) and radius 5 , the circle C with center B (4 ; 1) and radius 3 and point C with coordinates C (1 ; 1) . 1 a Show that the point C belongs to the circles C and C . b Show that the point D 56 29 ; − 34 29 is the second point of intersection of the circles C and C . 2 Justify that the triangle ABC is not a right-angled tri-angle. E.8365 In the plane provided with a O I ; J , consider the curve C f representative of the square function, the line with equation y = − 1 4 and the point F 0 ; 1 4 . Show that any point M of the curve C f is equidistant from the point F and the line (Δ) . Hint : note P the orthogonal project of the point M onto the straight line Δ . 7. Introduction to the middle of a segment https://chingmath.fr chapExoCorrec/11531 sacados/11531 chapExoCorrec/11534 sacados/11534 chapExoCorrec/9600 sacados/9600 chapExoCorrec/2705 sacados/2705 -2246I-6-4-22JO chapExoCorrec/10623 sacados/10623 chapExoCorrec/8039 sacados/8039 chapExoCorrec/8108 sacados/8108 chapExoCorrec/8365 sacados/8365 IJOCfFMP
IJOABCIxAxBx -3-2-1234I-3-2-123456JOABCD E.489 Consider the plane provided with the datum ( O ; I ; J ) and two points A and B with coordinates ( x A ; y A ) and ( x B ; y B ) respectively. Let I be the midpoint of segment [ AB ] . On the abscissa axis, the point of abscissa x is the midpoint of the two points of abscissa x A and x B . The aim of this exercise is to determine the coordinates of the point I as a function of the coordinates of those of the points A and B . 1 a Justify the following equality: x − x A = x B − x b From the previous equality, deduce the value of x as a function of x A and x B . 2 Justify that point I has abscissa x A + x B 2 . 3 Deduce that the point I has the following coordinates : I x A + x B 2 ; y A + y B 2 8. Middle of a segment E.2707 Proposition: In a plane equipped with an orthonormal coordinate system, consider the two points A ( x A ; y A ) and B ( x B ; y B ) . The midpoint M of the segment [ AB ] has co-ordinates : M x A + x B 2 ; y A + y B 2 Consider the plane equipped with the orthonormal coordi-nate system ( O ; I ; J ) and the four points A , B , C , and D indicated below : 1 Give the coordinates of points A , B , and C . 2 a Let K be the midpoint of segment [ AC ] . Determine the coordinates of K . b Let L be the midpoint of [ BD ] , determine the coordi-nates of point L . 3 Deduce the nature of quadrilateral ABCD . E.11499 Proposition: Let A ( x A ; y A ) and B ( x B ; y B ) be two points on the plane equipped with an orthonormal coor-dinate system. The midpoint M of the segment [ AB ] has coordinates : M = x A + x B 2 ; y A + y B 2 Consider the plane equipped with a coordinate system. 1 We have the coordinates : A ( − 3 ; 2) ; B (4 ; − 4) Determine the coordinates of point M , the midpoint of segment [ AB ] . 2 We have the coordinates : C (3 ; 1) ; D (5 ; − 1) Determine the coordinates of point N , the midpoint of segment [ CD ] . E.10624 In the plane with a coordinate system, consider the four points : A ( − 3 ; 4) ; B (6 ; 1) ; C (1 ; 0) ; D ( − 8 ; 3) 1 Determine the coordinates of points M and N , the mid-points of segments [ AC ] and [ BD ] , respectively. 2 Justify that the quadrilateral ABCD is a parallelogram. E.11532 In the plane equipped with a coordinate system, consider the four points : A (2 ; 1) ; B ( − 3 ; − 1) ; C ( − 8 ; − 5) ; D ( − 3 ; − 3) 1 Determine the coordinates of points M and N , the mid-points of segments [ AC ] and [ BD ] , respectively. 2 Justify that the quadrilateral ABCD is a parallelogram. E.9602 In the plane provided with an or-thonormal reference frame, consider the points : A 1 2 ; − 3 2 ; B ( − 2 ; 0) ; C − 1 3 ; 15 7 ; D 13 6 ; 9 14 Establish that the quadrilateral ABCD is a parallelogram. https://chingmath.fr chapExoCorrec/489 sacados/489 IJOABCIxAxBx chapExoCorrec/2707 sacados/2707 -3-2-1234I-3-2-123456JOABCD chapExoCorrec/11499 sacados/11499 chapExoCorrec/10624 sacados/10624 chapExoCorrec/11532 sacados/11532 chapExoCorrec/9602 sacados/9602
IJO 9. Length and middle E.8038 In the plane equipped with an or-thonormal coordinate system O ; I ; J , consider the circle C and two points A (2 ; 1) and B (10 ; 7) diametrically opposite on the circle C . Determine the coordinates of the point M , the center of the circle C , and the length of the circle’s radius. E.8072 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the three points : A (2.4 ; 2) ; B (1.4 ; − 3) ; C (3 ; − 2.8) 1 Determine the measure of segment [ AB ] . 2 The following measures are assumed : BC = 2.6 ; AC = 23.4 Justify that the triangle ABC is a right-angled triangle. 3 Consider the point D (0.8 ; 1.8) . Show that the quadrilat-eral ACBD is a rectangle. E.8061 In a reference frame O ; I ; J or-thonormal, consider the two points : A (1 ; 2) ; B (1 ; − 3) ; C (3.4 ; 0.2) 1 Justify that the line ( AB ) is parallel to the ordinate axis. The following measurements are assumed : AB =5 ; AC =3 2 Establish that the triangle ABC is right-angled at C . 3 Determine the coordinates of the middle of segment [ BC ] . E.8426 In a reference frame O ; I ; J or-thonormal, consider the three points : A (5 ; − 2) ; B ( − 1 ; 3) ; C (2.2 ; 4.9) 1 Determine the length of segment [ AB ] . 2 Determine the coordinates of the middle K of the seg-ment [ BC ] . E.2709 Consider the following four points characterized by their coordinates in an orthonormal ( O ; I ; J ) reference frame : A ( − 4 ; − 1) ; B ( − 3 ; − 4) ; C (3 ; − 2) ; D (2 ; 1) Show that the quadrilateral ABCD is a rectangle. E.950 Consider an orthonormal ref-erence frame ( O ; I ; J ) . The unit chosen is the centimeter . 1 Place points : A (2 ; 2) ; B ( − 4 ; 5) ; C ( − 4 ; − 2) 2 a Show that AC is equal to 52 cm b Is the ABC triangle isosceles at C ? Justify. 3 a Determine the coordinates of the middle K of [ AB ] . is b line ( CK ) the perpendicular bisector of segment [ AB ] ? Justify. E.923 1 In an orthonormal reference frame ( O ; I ; J ) , place the points : A ( − 3 ; 1) ; B − 3 2 ; 5 2 ; C (3 ; − 2) ; D 3 2 ; − 7 2 2 Show that : AC = 45 . 3 Prove that ABC is a right triangle at B . 4 Establish that the quadrilateral ABCD is a rectangle. 10. Middle and search for the coordinates of a point E.917 Consider the plane provided with an orthonormal reference frame ( O ; I ; J ) and the following points determined by their coordinates : A ( − 4 ; 1) ; B (1 ; 3) ; C ( − 2 ; − 3) . 1 Place on the above marker the points A , B , C . 2 Determine the coordinates of point K midpoint of seg-ment [ AC ] . 3 Let’s find the coordinates of the point D ( x D ; y D ) so that the quadrilateral ABCD is a parallelogram: a Justify that the coordinates of the point D must verify the following two equalities: 1 + x D 2 = − 3 ; 3 + y D 2 = − 1 b Deduce from the following equalities the coordinates of the point D ; then place this point in the reference frame. https://chingmath.fr chapExoCorrec/8038 sacados/8038 chapExoCorrec/8072 sacados/8072 chapExoCorrec/8061 sacados/8061 chapExoCorrec/8426 sacados/8426 chapExoCorrec/2709 sacados/2709 chapExoCorrec/950 sacados/950 Groupe Ouest - Juin 2004 - 3,5 points chapExoCorrec/923 sacados/923 chapExoCorrec/917 sacados/917 IJO
r569-0 -4-3-2-1234I-123JO E.511 In a reference frame O ; I ; J of the plane, consider the points : A (3 ; 1) ; B ( − 4 ; 2) ; C ( − 1 ; 4) 1 Consider the point D symmetrical to the point C with respect to the point B . Determine the coordinates of the point D . 2 Let E be the point in the plane such that the segments [ AC ] and [ BE ] have the same midpoint. Determine the coordinates of the point E . E.11533 In a coordinate system O ; I ; J , consider three points A (5 ; 2 ; 3 ; 1) , B ( − 3 ; 5 ; 4 ; 7) , C . Determine the coordinates of point C so that point B is the midpoint of segment [ AC ] . Hint: Keep track of your research and calculations. E.11590 In a coordinate system O ; I ; J , consider three points A ( − 3 ; 2 ; 1 ; 7) , B (9 ; 5 ; − 4 ; 2) , C . Determine the coordinates of point C so that point B is the midpoint of segment [ AC ] . Hint: Keep track of your research and calculations. E.522 In an ( O ; I ; J ) orthonormal coor-dinate system, consider the three points A , B and C with coordinates : A (2 ; 1) ; B ( − 3 ; − 1) ; C (1 ; − 3) 1 Determine the measure of length AB . 2 Let K be the midpoint of segment [ AC ] . Determine the coordinates of point K . 3 Determine the coordinates of the point D such that the quadrilateral ABCD is a parallelogram. E.9622 In the plane provided with a reference frame O ; I ; J , consider the three points : A (3.25 ; − 1.5) ; B (1 ; 1.5) ; C (4 ; − 0.75) 1 Show that the triangle ABC is isosceles at B . 2 We admit that the segment [ AC ] admits for miilieu the point I (3.625 ; − 1.125) a Determine the coordinates of the point D so that the segment [ BD ] admits the point I as its midpoint. b Deduce the natude of the quadrilateral ABCD justify-ing your answer. 11. Relations between quadrilaterals E.4602 Note: relationships between particular quadrilaterals are available in the attached doc-ument : Consider the plane provided with an orthonormal reference frame O ; I ; J and the points : A ( − 2 ; 3) ; B (4 ; 5) ; D ( − 1 ; 0) 1 Determine the coordinates of the single point C of the plane so that the quadrilateral ABCD is a parallelogram. 2 Demonstrate that the quadrilateral ABCD is a rectan-gle. E.9603 Consider the plane provided with an orthonormal reference frame O ; I ; J and the points : A ( − 2 ; 3) ; B (4 ; 5) ; E (2 ; 1) ; F (0 ; 7) 1 Demonstrate that the quadrilateral AEBF is a parallel-ogram. 2 Demonstrate that the parallelogram AEBF is a rhom-bus. 3 Demonstrate that the rhombus AEBF is a square. E.9621 In the plane provided with a refer-ence frame O ; I ; J orthonormal, consider the four points A , B , C , D whose coordinates are: A ( − 2 ; 1) ; B (2.5 ; − 0.5) ; C (3 ; 1) ; D ( − 1.5 ; 2.5) We assume that : AC = 5 . 1 Show that the middle I of segment [ BD ] has coordinates : I (0.5 ; 1) 2 Justify that the quadrilateral ABCD is a parallelogram. 3 Demonstrate that the parallelogram ABCD is a rectan-gle. Note: the marker below may help you check your results : https://chingmath.fr chapExoCorrec/511 sacados/511 chapExoCorrec/11533 sacados/11533 chapExoCorrec/11590 sacados/11590 chapExoCorrec/522 sacados/522 chapExoCorrec/9622 sacados/9622 chapExoCorrec/4602 sacados/4602 r569-0 chapExoCorrec/9603 sacados/9603 chapExoCorrec/9621 sacados/9621 -4-3-2-1234I-123JO
-4-2246810I-2246JOABC E.4593 Consider the plane provided with a O ; I ; J orthonormal coordinate system. Consider the four points : A (3 ; 2) ; B (9 ; 5) ; C (1 ; 6) 1 Determine the coordinates of the point D so that the quadrilateral ABCD is a parallelogram. 2 Consider the point E (7 ; 9) . Show that the quadrilateral ABEC is a rectangle. E.4814 The plane is given a reference frame O ; I ; J . Then consider the two points A , B and the vector −→ u defined by: A (0 ; − 4) ; B (2 ; 4) ; −→ u ( − 6 ; 10) We define the point C as the image of the point A by the translation of the vector −→ u . 1 Justify that the point C has coordinates ( − 6 ; 6) . 2 Determine the coordinates of the point D such that ABCD is a parallelogram. We accept the measures : AB =2 17 ; AC =2 34 3 Determine the nature of the quadrilateral ABCD . 12. Locating and vectoring: analytical geometry E.944 In an orthonormal coordinate system ( O ; I ; J ) with units of centimeters, plot the following points : A (6 ; 5) ; B (2 ; − 3) ; C ( − 4 ; 0) 1 Show that : AB = 80 . The following additional information is given : AC = 125 ; BC = 45 . 2 Determine the type of triangle ABC . Justify your an-swer. 3 Calculate the area of triangle ABC in cm 2 . 4 Consider the circumscribed circle of triangle ABC . a Specify the position of its center, labeled K , and the length of its radius. Justify your answer. Label K . b Calculate the coordinates of K . 5 a Calculate the coordinates of the vector −→ AC . b From this, determine the coordinates of the point D such that ACBD is a parallelogram. c Plot the point D . E.948 The plane is equipped with an orthonormal coordinate system O ; I ; J . Consider the coordinates of the following points : A ( − 4 ; 3) ; B ( − 1 ; − 1) ; C (7 ; 5) 1 Determine the coordinates of the vector −−→ AB , then calcu-late the length of the segment [ AB ] . For the rest of the problem, we will assume that : BC = 10 ; AC = 125 2 Prove that triangle ABC is a right triangle. 3 Find the coordinates of the midpoint M of [ AC ] and plot point M on the figure above. 4 Prove that : MB = MC . 5 Let N be the image of point M under the translation by vector −−→ AB . a Which vector relationship does point N satisfy? b Use this to calculate the coordinates of point N . c Plot point N on the coordinate plane. 6 Prove that vectors −−→ AM , −−→ BN , and −−→ MC are equal. 7 Prove that the quadrilateral BMCN is a rhombus. 8 Prove that triangle ABC and rhombus BMCN have the same area. https://chingmath.fr chapExoCorrec/4593 sacados/4593 chapExoCorrec/4814 sacados/4814 chapExoCorrec/944 sacados/944 chapExoCorrec/948 sacados/948 -4-2246810I-2246JOABC
-4-22468I-4-224JO E.926 Consider a plane equipped with an orthonormal coordinate system ( O ; I ; J ) in which the unit is the centimeter. 1 Draw such a coordinate system and, throughout the ex-ercise, complete your diagram. 2 Plot the points : M (1 ; 3) ; N ( − 1 ; 5) ; P ( − 3 ; 1) 3 Establish the following equalities: MN = 8 ; NP = MP = 20 . 4 Deduce the type of triangle MNP . 5 Let A be the midpoint of [ MN ] . Show, without calcula-tion, that triangle APN is a right triangle. 6 Calculate the coordinates of A . 7 Construct point R such that : −−→ MR = −−→ PN 8 Calculate the coordinates of the vector −−→ PN . 9 Determine the coordinates of point R from the questions 6 et 7 . E.945 Consider an orthonormal co-ordinate system ( O ; I ; J ) , whose representation is shown be-low : Consider the following three points : A ( − 4 ; 3) ; B (3 ; 2) ; C (1 ; − 2) Part A 1 Plot the points A , B , and C in the coordinate system ( O ; I ; J ) . 2 a Calculate AB . b Suppose the calculation yields: AC = 50 ; BC = 20 . What can we deduce from this about triangle ABC ? 3 Let H be the midpoint of segment [ BC ] . Verify by cal-culation that H has coordinates (2 ; 0) . 4 Justify that the line ( AH ) is a height of triangle ABC . 5 a Prove that : AH = 45 . b Calculate the area of triangle ABC Part B 1 Calculate the coordinates of vector −→ AC . 2 Point D is the image of point B under the translation by vector −→ AC . a Locate point D . b Show by calculation that D has coordinates (8 ; − 3) . 3 What is the type of quadrilateral ACDB ? Justify your answer. E.916 No graphical representation is required to solve this problem. In the plane with an orthonormal coordinate system ( O ; I ; J ) , consider the points : A ( − 3 ; 2) ; B (0 ; 4) ; C ( − 1 ; − 1) . 1 a Find the coordinates of the vector −−→ AB b Determine the coordinates of the point D such that the quadrilateral ABDC is a parallelogram. 2 a Find the length of segment [ AB ] ? b Find the coordinates of the midpoint I of segment [ AB ] . 13. Use of non-rational coordinates E.9623 In the plane provided with an orthonormal reference frame O ; I ; J , consider the three points : A 3 − 2 ; 3 − 17 4 ; B 3+4 ; 3+ 1 4 C 1 − 8 3 ; − 2 + 13 3 We admit that the triangle ABC is isosceles in C . 1 Determine the length of segment [ AB ] . 2 Determine the coordinates of the middle of the segment [ AB ] . 3 Show that the area A of the triangle ABC has the value : A = 225 4 · 3 E.2740 In the plane provided with an orthonormal reference frame, consider the following three points : A ( − 5 ; − 4) ; B (3 ; − 2) ; C − 3 − 1 ; 4 3 − 3 Show that the triangle ABC is equilateral. E.2754 Consider the plane provided with the reference frame ( O ; I ; J ) orthonormal. Consider the fol-lowing three points : A − 5 3 ; 1 ; B ( − 2 ; 1) ; C − 11 6 ; 1 − 3 6 1 Show that : AC = 1 3 . 2 Show that ABC is an equilateral triangle. https://chingmath.fr chapExoCorrec/926 sacados/926 chapExoCorrec/945 sacados/945 -4-22468I-4-224JO chapExoCorrec/916 sacados/916 dddddd chapExoCorrec/9623 sacados/9623 chapExoCorrec/2740 sacados/2740 chapExoCorrec/2754 sacados/2754
-2-12345I-2-1234JOABCC -2-1234I-2-1234JOABC E.2753 Consider the plane provided with the reference frame O ; I ; J orthonormal. Consider the fol-lowing three points : A ( − 5 ; 2) ; B (1 ; 0) ; C − 3 − 2 ; 1 − 3 3 1 Establish equality: AC = 40 . 2 Determine the nature of the triangle ABC . E.8040 Consider the plane with a reference frame O ; I ; J and the circle C with center A ( − 3 ; 1) and radius 2 . Of the points B − 7 5 ; 11 5 and C − 2 ; 3+1 , give the point(s) belonging to the circle C . 14. Coordinate search and remarkable identity E.946 We equip the plane with an orthonor-mal basis O ; I ; J . Consider the circle C with center A (1 ; 1) and diameter 5 . The points B and C are the points of intersection of the circle C with the line of equation x =3 . 1 Give the x-coordinates of points C and B . 2 Justify that the y-coordinate y C of point C satisfies the following equality: 2 2 +(1 − y C ) 2 =6 ; 25 3 Recall the following proposition : Proposition: if the squares of two numbers are equal, then these two numbers are either equal or opposite. Which translates to : x 2 = y 2 = ⇒ x = y ou x = − y Deduce the coordinates of points C and B . E.2724 The plane is given an orthonormal reference frame O ; I ; J . Consider the two points C and B of coordinates (1 ; 1) and 11 5 ; 13 5 respectively and the circle C with center C and passing through point B . Consider the line parallel to the y-axis and passing through the point B ; it intercepts the circle C a second time at the point A . The aim of this exercise is to determine the coordinates of the point A . 1 Determine the measure of the radius of the circle C . 2 Establish that the ordinate of the point A verifies the equation : 6 5 2 + y A − 1 2 = 4 3 Deduce the value of the ordinate of point A . https://chingmath.fr chapExoCorrec/2753 sacados/2753 chapExoCorrec/8040 sacados/8040 chapExoCorrec/946 sacados/946 -2-12345I-2-1234JOABCC chapExoCorrec/2724 sacados/2724 -2-1234I-2-1234JOABC
-4-3-2-1234I-2-1234JO E.4617 In the plane provided with the refer-ence frame O ; I ; J orthonormal, consider the points A and B : A (3 ; 1) ; B 3 5 ; − 4 5 Consider the circle C of center A and radius 3 . We’ll complete the landmark as the questions progress. 1 Justify that the point B belongs to the circle C . 2 Determine the coordinates of the points on the circle C having 6 5 as their abscissa. 3 The coordinates of the center of gravity G of a triangle ABC are given by the formula : G x A + x B + x C 3 ; y A + y B + y C 3 Determine the coordinates of the point C so that the triangle ABC admits the point J as its center of gravity. E.2741 The plane is provided with an or-thonormal coordinate system ( O ; I ; J ) . Consider the follow-ing three points defined by their coordinates : A ( − 5 ; − 3) ; B (1 ; − 5) ; D ( − 1 ; 1) 1 Determine the measure of segment [ AB ] . 2 Determine the coordinates of point K midpoint of seg-ment [ AD ] . 3 Let C be a point such that the triangle ABC is equilat-eral: a Justify that the point C verifies each of these two equa-tions : x C +5 2 + y C +3 2 = 40 ; x C − 1 2 + y C +5 2 = 40 b Assuming that the point C has abscissa − 3 − 2 , jus-tify that its ordinate y C verifies the following system of equations : y C 2 + 6 y C + 21 − 6 3 = 40 y C 2 + 10 y C + 37 + 6 6 = 40 c Deduce the coordinates of point C . 4 Determine the coordinates of the point M such that the quadrilateral ABDM is a parallelogram. 15. Unclassified exercises E.942 For each line in the table below, three answers are provided, identified by the numbers 1. , 2. , 3. . Only one is correct. Write the number corresponding to the correct answer in the right-hand column. All questions are independent. Answer 1. Answer 2. Answer 3. A. If A (5; 1) and B (2; 3) , then −−→ AB has coordinates : (3; − 4) (7; 2) ( − 3; 2) B. If A (5; − 1) and B (2; 3) are in an orthonormal coor-dinate system, then AB is equal to : 5 1 7 C. If D is the image of E under the vector translation −−→ MN , then : −−→ MN = −−→ DE −−→ ED = −−→ MN −−→ ED = −−→ NM D. If RSTU is a parallelo-gram, then −→ RS + −−→ RU is equal to : −→ TR −→ SU −→ RT E.915 1 In an orthonormal frame of reference O ; I ; J whose unit is the centimeter, place the following three points : A (6 ; 0) ; L (0 ; 8) ; K (4 ; 10) 2 Calculate length AL . 3 We give : AK = 104 and LK = 20 . Show that the triangle AKL is not right-angled at L . 4 a Construct the point L , symmetrical to L with re-spect to the height from A of the triangle AKL . b Deduce the length AL . c Determine approximately (by graphical reading) the coordinates of L . 5 We admit that, if x is the abscissa of a point M of the line ( LK ) then the ordinate of M is 1 2 x +8 : a Establish the equality below : AM 2 = 5 4 x 2 − 4 x + 100 b Deduce the values of x for which, we have : AM = 10 . c What are then the exact coordinates of L ? https://chingmath.fr chapExoCorrec/4617 sacados/4617 -4-3-2-1234I-2-1234JO chapExoCorrec/2741 sacados/2741 chapExoCorrec/942 sacados/942 chapExoCorrec/915 sacados/915
-3-2-12345I-1234JO -5-4-3-2-12345I-2-1234JO E.927 The plane is equipped with an orthonormal coordinate system O ; I ; J , as shown below : 1 a Plot the following two points : A ( − 2 ; 1) ; B (1 ; 2) b Determine the coordinates of vector −−→ AB graphically. 2 a Plot the points R and C , which are the respective images of points O and B under the translation by vector −−→ AB . b Find the coordinates of points R and C . 3 List two vectors equal to −−→ AB . Prove that BCRO is a parallelogram. 4 Copy and complete the following equations without proof : −→ OA + −−→ AB = : : : ; −−→ CB + −→ CR = : : : 5 Let K be the center of the parallelogram BCRO . Calcu-late the coordinates of K . E.6690 The plane is provided with an or-thonormal reference frame. Consider the points A ( − 2.5 ; 0.5) , B ( − 1.5 ; 2.5) and C (0.5 ; − 1) . 1 Place the points A , B and C in the marker below. 2 Determine, by calculation, the coordinates of the vectors −−→ AB and −→ AC . 3 Place the point D such that : −−→ AD = −−→ AB + −→ AC (We’ll show the construction lines) 4 a Give the coordinates of the vector obtained by the sum : −−→ AB + −→ AC . b Deduce, by calculation, the coordinates of the point D . For the rest, we assume that D (1.5 ; 1) . 5 a Determine the coordinates of the vector −−→ CD . b Deduce that the quadrilateral ABDC is a parallelo-gram. is 6 ABDC a rectangle? Justify. 7 We give E − 3 4 ; 4 . Are the points A , B and E aligned? E.4736 In the plane provided with a refer-ence frame O ; I ; J , the straight line ( d ) of slope-intercept form : ( d ) : y = − 3 x − 1 Determine the coordinates of a point M on the line ( d ) such that the triangle IJM is rectangular at J . https://chingmath.fr chapExoCorrec/927 sacados/927 -3-2-12345I-1234JO chapExoCorrec/6690 sacados/6690 -5-4-3-2-12345I-2-1234JO chapExoCorrec/4736 sacados/4736
x-4-3-2-1012y-1123AB(d xy11ABC E.6685 The plane is provided with an orthonormal coordinate system O ; I ; J . Consider the straight line ( d ) shown below : Let ( d ) be the straight line passing through the points A and B having coordinates : A ( − 3 ; 2) ; B − 1 ; 3 2 1 Determine the expression of the linear function f . 2 Consider the linear function g defined by: g ( x ) = 4 · x − 3 Let (Δ) be the representative curve of the function g . a Justify that the point with coordinates C 1 2 ; − 1 be-longs to the line (Δ) . b Determine the coordinates of the point M of intersec-tion of the straight lines ( d ) and (Δ) . c Draw the representative curve of the function g . 3 Establish that the triangle AMC is a right-angled trian-gle in M . E.6694 In the plane provided with a ref-erence frame O ; I ; J , consider the three points A , B and C with coordinates : A ( - 1 ; 1) ; B (3 ; - 2) ; C ( - 1 ; - 4) 1 Demonstrate that the trian-gle ABC is an isosceles tri-angle at A . 2 Determine, by calculation, the slope-intercept formof the line ( AB ) . 3 Let ( d ) be the line with slope-intercept form ( d ) : y = − 2 x − 1 a Determine the coordinates of point K midpoint of seg-ment [ BC ] . b Demonstrate that the straight line ( d ) is the bisector of the angle ∠ CAB . E.8218 Consider the two points G (1 ; 2) and H (4 ; 1) and the line ( d ) of equation : ( d ) : y = 3 x − 6 Show that line ( d ) is the perpendicular bisector of segment [ GH ] . E.9797 Consider the plane provided with a ref-erence frame O ; I ; J orthonormal and Consider the two points K (3 ; 3) and L (6 ; 1) and the circle C of diameter [ KL ] . The straight line (Δ) has equation : (Δ) : y = x − 2 1 Expand expression : 2( x − 3)(2 x − 11) . 2 Let M be a point on the line (Δ) . Determine the coordi-nates of the various points M of (Δ) making the triangle KLM rectangular at M . https://chingmath.fr chapExoCorrec/6685 sacados/6685 x-4-3-2-1012y-1123AB(d chapExoCorrec/6694 sacados/6694 xy11ABC chapExoCorrec/8218 sacados/8218 chapExoCorrec/9797 sacados/9797