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 notationλ→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 the
points A(−3,3) ; B(2,3) ; C(2,1) ; D(−3,1).
b. What problem are we facing in this situation ?
c. Add a conditional instruction to avoid that problem.
4. a. Apply this algorithm to the points A(−2; 2) ; B(1; 3) ; C(7; 5) ;D(4; 4).
b. Place the points in a Cartesian coordinate system and check the output of the algorithm. Com- ment the result.
5. This algorithm is not always work- ing. Find some input points for which there is a problem.
begin
Input: (xA, yA), (xB, yB), (xC, yC), (xD, yD) coordinates of four points;
xB−xA →a ; yB−yA→b ; xC−xD →a′ ; yC −yD →b′ ;
a
a′ →r1 ;
b
b′ →r2 ;
if r1 =r2 then
Output: “ABCD is a parallelogram” ; else
Output: “ABCD is not a parallelogram” ; end if
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) andU(5,4).
1. Find the coordinates of vectors −ET ,−−−→ −EB,−−−−→ −U E−−−−→and −BU−−−−→. 2. a. Compute the lengths ET and EB.
b. What type of quadrilateral is T U BE? Explain.
c. Find the coordinates of F, centre of T U BE, and plot it.
3. Let (C) be the circle centered atEpassing throughB, andAthe second intersection point of that circle with the x-axis.
a. What type of triangle is AT B? Explain.
b. Prove that the lines (AT) and (EF) are parallel.
c. Compare the lengths EF and AT.
4. What is the image of triangleAT E under the translation that shifts A toE? 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 are unknown 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 are equal.
8.5 Let →u −2 5
!
and →v 3
1 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 respective coordinates (−2,−1), (0,1) and (−4,0), and another pointDwith 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 LetA(−2,−1),B(3,2) andC(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−−−−→ .
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 ofx 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 coordinate system.
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′) are collinear if and only if their coordinates . . . , that is if . . . .
8.9 Using the property discovered in the previous exercise, decide whether the two vectors 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 coordinate system (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 ofI.
b. Compute the coordinates ofJ and K.
c. Prove thatI, J and K are collinear.
8.11
Í
The aim of the algorithm on the right-hand side is to decide if three points are collinear or not.1. Copy and fill the conditional instruc- tion.
2. Apply the algorithm to the points A(−2;−1) ; B(1; 1) ; C(7; 5).
3. Apply the algorithm to the points A(−2; 5) ; B(2; 2) ; C(5; 0).
4. What instruction should you add at the beginning of the algorithm to make sure that the three points are distinct ?
begin
Input: (xA, yA), (xB, yB), (xC, yC) coordinates of three 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 are two 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 BIJd .
b. Prove thatDIJ[ = 180o. Conclude.
2. Method 2 : In a coordinate system Use a convenient coordinate system to prove that 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√23.
8.13 LetABC be a triangle with AB = 8 cm,AC = 6 cm andBC = 5 cm. We consider the points M and N such that : −CN−−−−→=−1
4
−−−−−→
AC and −AM−−−−−→= 34−AB−−−−→. 1. Place the points M and N.
2. a. Write −BC−−−−→ as a linear combination of−AB−−−−→ and −AC.−−−−→ b. Write −M N−−−−−−→as a linear combination of −AB−−−−→and −AC.−−−−→ c. Deduce that the lineM N is parallel to the line BC.
3. a. Give the coordinates of pointsA,B,C,M,N in the coordinate system (A, B, C).
b. Compute the coordinates of−M N−−−−−−→ and −BC−−−−→.
c. Use the vectors to prove that the line M N 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
1. a. Why is (A, B, C) a coordinate system of the plane ? b. Give the coordinates of the pointsA,B, C, M and N. 2. Prove that the lines M N and BC are parallel one to another :
a. first, using the coordinates of−M N−−−−−−→ and −BC−−−−→;
b. second, decomposing −M N−−−−−−→ and −BC−−−−→as linear combinations of−AB−−−−→ and −AC.−−−−→ 3. a. What are the coordinates of the pointP such thatM N P B is a parallelogram ?
b. Prove thatP is on the lineBC.
8.15 LetABC be a scalene, non-flat, triangle. LetM be the midpoint of segment [AB] and N the midpoint of segment [AC]. We call K and L the respective midpoints of the segments [CM] and [BN]. Let P and Qbe the points defined by the relations :
−−−−−→
N P = 1 3
−−−−−−−→
N M and −M Q−−−−−−→= 1 3
−−−−−−−→
M N . 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 coordinate system.
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, are collinear.
8.16 In a Cartesian coordinate system (O;I, J), with unit 1 cm, consider the points A(−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 pointK such that ABHK is a parallelogram.
b. Compute the coordinates of the point Lsymmetric of H around A.
c. Compute the coordinates of the point C such that −HC−−−−→ =−2−HB.−−−−−→ d. Are the pointsC, K and L collinear ?
3. LetI be the midpoint of [AB] and J the midpoint of [AC]. Let Gbe the point such that −GA−−−−→+−GB−−−−→+−GC−−−−→=→0
a. Compute the coordinates of the pointsI,J and G.
b. Prove thatC,GandIare collinear, and then thatB,GandJare 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 pointM.
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). Compute the values of xE and yF.
