Section européenne Plane geometry 3

Vectors and coordinates

A la fin de ce chapitre, vous devez être capable de :

• connaître les coordonnées (xB − xA, yB − yA) d’un vecteur−−−→AB ;

• calculer les coordonnées de la somme de deux vecteurs dans un repère ;• utiliser la notation λ

→u ;

• établir la colinéarité de deux vecteurs.

Vector coordinates

8.1 In a Cartesian coordinate system, consider the points :

A(1, −3) ; B(4, −1) ; C(2, 1) and D(−1, −1)

1. Show that quadrilateral ABCD is a parallelogram :

a. using lengths ;

b. using the coordinates of the midpoints of the diagonals ;

c. using the coordinates of the vectors−−−−−→AB and

−−−−−→DC.

2. Same question with the points A(1, 2), B(1, −1), C(−1, −2) and D(−1, 1).

8.2 Í Consider the algorithm below :

1. Apply the algorithm in the following situations :

a. A(−4, 3) ; B(1, 5) ; C(3, 2) ; D(−2, 0).

b. A(−3, 3) ; B(0, 5) ; C(3, 2) ; D(−2, 1).

2. What is the aim of this algorithm ?

3. a. Apply this algorithm to thepoints A(−3, 3) ; B(2, 3) ;C(2, 1) ; D(−3, 1).

b. What problem are we facing inthis situation ?

c. Add a conditional instruction toavoid that problem.

4. a. Apply this algorithm to thepoints A(−2; 2) ; B(1; 3) ;C(7; 5) ; D(4; 4).

b. Place the points in a Cartesiancoordinate system and check theoutput of the algorithm. Com-ment the result.

5. This algorithm is not always work-ing. Find some input points for whichthere is a problem.

beginInput: (xA, yA), (xB, yB),

(xC , yC), (xD, yD)coordinates of fourpoints;

xB − xA → a ;yB − yA → b ;xC − xD → a′ ;yC − yD → b′ ;a

a′→ r1 ;

b

b′→ r2 ;

if r1 = r2 thenOutput: “ABCD is a

parallelogram” ;

elseOutput: “ABCD is not a

parallelogram” ;

end if

end

8.3 In a Cartesian coordinate system (O; I, J) with unit 1 cm, we consider the points :

E(−3, 0) ; B(2, 0) ; T (0, 4) and U(5, 4).

1. Find the coordinates of vectors−−−−→ET ,

−−−−−→EB,

−−−−−→UE and

−−−−−→BU .

2. a. Compute the lengths ET and EB.

b. What type of quadrilateral is TUBE ? Explain.

c. Find the coordinates of F , centre of TUBE, and plot it.

3. Let (C ) be the circle centered at E passing through B, and A the second intersectionpoint of that circle with the x-axis.

a. What type of triangle is ATB ? Explain.

b. Prove that the lines (AT ) and (EF ) are parallel.

c. Compare the lengths EF and AT .

4. What is the image of triangle ATE under the translation that shifts A to E ?

8.4 Let A(2, 3), B(1, −1), C(−3, −3) be three points in a coordinate system (O; I, J)and D a fourth point such that ABCD is a parallelogram. The coordinates of D areunknown and will be denoted (a, b).

1. Compute the coordinates of vector−−−−−→AB.

2. Write the coordinates of−−−−−→DC using the unknowns a and b.

3. Find out the values of a and b.

4. Compute the lengths AB, BC, CD and DA and check that the opposite sides areequal.

8.5 Let →u

(−25

)and →

v

(31

4

)be two vectors defined by their coordinates in a Cartesian

coordinate system (O; I, J). Compute the coordinates of the following vectors :

→u + →

v , 2→u , −

3

7→v , −3→

u + →v , 2→

v + 0, 8→u , 1, 5→

u −

3

5→v .

8.6 In a coordinate system (O; I, J), consider the points A, B and C with respectivecoordinates (−2, −1), (0, 1) and (−4, 0), and another point D with coordinates (x, y) such

that−−−−−→CD = 2

−−−−−→AB.

1. Plot the point D.

2. Find the coordinates of vectors−−−−−→AB and

−−−−−→CD.

3. Compute the coordinates of point D and check your result with the graph.

8.7 Let A(−2, −1), B(3, 2) and C(1, 5) be three points in a coordinate system (O; I, J).

1. Compute the coordinates of vectors−−−−−→AB and

−−−−−→AC.

2. Compute the coordinates of the point I, midpoint of segment AB.

3. Compute the length AC.

4. Compute the coordinates of vectors−−−−−→AB +

−−−−−→AC, 3

−−−−−→AB and 2

−−−−−→AB + 3

−−−−−→AC .

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5. Let K be the point with coordinates (x, y), such that−−−−−−→CK = 3

−−−−−→AB.

a. Using x and y, give the coordinates of the vector−−−−−−→CK.

b. Find out the values of x and y.

6. Compute the coordinates of the point L defined by−−−−→AL + 2

−−−−→BL +

−−−−→CL =

→0.

Collinearity and coordinates

8.8 Let A(−2, 1), B(2, 3), C(0, −2), D(2, −1) be four points in a Cartesian coordinatesystem.

1. Plot the points A, B, C, D.

2. a. What can you say about the vectors−−−−−→AB and

−−−−−→CD ?

b. Find the coordinates of vectors−−−−−→AB and

−−−−−→CD, denoted respectively (x, y) and

(x′, y′). Are these two couples of coordinates proportional ?

c. Compute xy′− yx′.

3. a. Do the vectors−−−−−→AC and

−−−−−→BD look collinear ?

b. Find the coordinates of vectors−−−−−→AC and

−−−−−→BD, denoted respectively (s, t) and

(s′, t′). Are these two couples of coordinates proportional ?

c. Compute st′− ts′.

4. Use the examples from the previous questions to copy and fill the property below.

Two vectors →u and →

v with respective coordinates (x, y) and (x′, y′) arecollinear if and only if their coordinates . . . , that is if . . . .

8.9 Using the property discovered in the previous exercise, decide whether the twovectors in each pair below are collinear or not.

1.→u(2, 5) and →

v (−4, −10) ;

2.→u(−3, 9) and →v (2, 6) ;

3.→u(5, 1) and →

v (−10, 2) ;

4.→u(−4, 12) and →

v (6, −18) ;

5.→u(0, 7) and →v (0, −8) ;

6.→u(0, 7) and →

v (−8, 0).

8.10 Let A(−7, 4), B(−4, 10), C(10, 13) and D(6, 5) be four points in a coordinatesystem (O; I, J).

1. a. Compute the coordinates of the vectors−−−−−→AB and

−−−−−→CD.

b. Deduce that the quadrilateral ABCD is a trapezium.

2. Let I be the point such that−−−→IA = 3

−−−−−→AD, J and K the midpoints of the segments

AB and CD.

a. Compute the coordinates of I.

b. Compute the coordinates of J and K.

c. Prove that I, J and K are collinear.

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8.11 Í The aim of the algorithm on theright-hand side is to decide if three pointsare collinear or not.

1. Copy and fill the conditional instruc-tion.

2. Apply the algorithm to the pointsA(−2; −1) ; B(1; 1) ; C(7; 5).

3. Apply the algorithm to the pointsA(−2; 5) ; B(2; 2) ; C(5; 0).

4. What instruction should you add atthe beginning of the algorithm tomake sure that the three points aredistinct ?

beginInput: (xA, yA), (xB, yB),

(xC , yC) coordinates ofthree distinct points;

xB − xA → a ;yB − yA → b ;xC − xA → a′ ;yC − yA → b′ ;a

a′→ r1 ;

b

b′→ r2 ;

if . . . then

Output: . . . ;else

Output: . . . ;end if

end

8.12 In the picture below, ABCD is a square with side one unit, AIB and BJC aretwo equilateral triangles. The aim of this exercise is to prove, by two different methods,that the points D, I, J are collinear.

×

A×

B

×C

×D

×I

× J

1. Method 1 : With geometric angles

a. Compute the measures of the angles

DIA, AIB and BIJ .

b. Prove that DIJ = 180o. Conclude.

2. Method 2 : In a coordinate system

Use a convenient coordinate system to provethat the points D, I and J are collinear.You may use the fact that in an equilateral

triangle with side a, the length of an altitude

is a√

3

2.

8.13 Let ABC be a triangle with AB = 8 cm, AC = 6 cm and BC = 5 cm. We consider

the points M and N such that :−−−−−→CN = −

1

4

−−−−−→AC and

−−−−−−→AM = 3

4

−−−−−→AB.

1. Place the points M and N .

2. a. Write−−−−−→BC as a linear combination of

−−−−−→AB and

−−−−−→AC.

b. Write−−−−−−−→MN as a linear combination of

−−−−−→AB and

−−−−−→AC.

c. Deduce that the line MN is parallel to the line BC.

3. a. Give the coordinates of points A, B, C, M , N in the coordinate system (A, B, C).

b. Compute the coordinates of−−−−−−−→MN and

−−−−−→BC .

c. Use the vectors to prove that the line MN is parallel to the line BC.

8.14 Let ABC be a non-flat triangle, M and N two points such that :

−−−−−−→AM =

3

5

−−−−−→AB and

−−−−−→AN =

2

5

−−−−−→AB +

1

5

−−−−−→AC

4

1. a. Why is (A, B, C) a coordinate system of the plane ?

b. Give the coordinates of the points A, B, C, M and N .

2. Prove that the lines MN and BC are parallel one to another :

a. first, using the coordinates of−−−−−−−→MN and

−−−−−→BC ;

b. second, decomposing−−−−−−−→MN and

−−−−−→BC as linear combinations of

−−−−−→AB and

−−−−−→AC.

3. a. What are the coordinates of the point P such that MNP B is a parallelogram ?

b. Prove that P is on the line BC.

8.15 Let ABC be a scalene, non-flat, triangle. Let M be the midpoint of segment [AB]and N the midpoint of segment [AC]. We call K and L the respective midpoints of thesegments [CM ] and [BN ]. Let P and Q be the points defined by the relations :

−−−−−→NP =

1

3

−−−−−−−→NM and

−−−−−−−→MQ =

1

3

−−−−−−−→MN.

1. a. Draw a picture.

b. Explain why (A; B, C) is a Cartesian coordinate system in the plane.

c. Give the coordinates of A, B, C, M , N , K, L, P and Q in this coordinatesystem.

2. Prove that the lines (P Q) and (KL) are parallel.

3. Prove that the points A, P , K on one hand and A, Q, L on the other hand, arecollinear.

8.16 In a Cartesian coordinate system (O; I, J), with unit 1 cm, consider the pointsA(−1, 0), B(1, −6) and H(3, −2).

1. Draw a picture and fill it all along the exercise.

2. a. Compute the coordinates of the point K such that ABHK is a parallelogram.

b. Compute the coordinates of the point L symmetric of H around A.

c. Compute the coordinates of the point C such that−−−−−→HC = −2

−−−−−−→HB.

d. Are the points C, K and L collinear ?

3. Let I be the midpoint of [AB] and J the midpoint of [AC]. Let G be the point such

that−−−−−→GA +

−−−−−→GB +

−−−−−→GC =

→0

a. Compute the coordinates of the points I, J and G.

b. Prove that C, G and I are collinear, and then that B, G and J are also collinear.

c. What is the point G in the triangle ABC ?

4. Let M be the point defined by the relation 3−−−−−−−→BM = 3

−−−−−→BA + 2

−−−−−→BC.

a. Compute the coordinates of point M .

b. Prove that the lines (AM) and (BC) are parallel.

5. a. Compute AH2, HB2 and AB2. What can you deduce about the lines (AH)and (BC) ?

b. Is the triangle ABC right-angled ? Compute the area of triangle ABC.

6. The line (LK) intersects the axes at two points E(xE , 0) and F (0, yF ). Computethe values of xE and yF .

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