System of reference in a plane

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Application of vectors to geometric problems II
Block:Geometry
 
2.2. LINEAR POINTS
2.2.1. PROOF THAT THREE POINTS ARE LINEAR

The points A(x1,y1), B(x2,y2), C(x3,y3) are always linear when the vectors AB and BC are in the same direction. This occurs when their coordinates are proportional: 

AB = (x2-x1 , y2-y1)   
BC = (x3-x2 , y3-y2)

1.- In this figure you can move the points B and C, to prove that the coordinates of the vectors AB and BC are proportional, and therefore that the points A, B and C are linear. 
Note in your exercise book the coordinates of A, B and C, those of the vectors AB and BC and the proportion between the x and the y values at he start of the figure. 

2.- Calculate the coordinates of BC if C=(5,2) and A and B do not change. 

3.- Now calculate the ratio between the x of AB and the x of BC.

4.- Also calculate the ratio between the y of AB and the y of BC. It should give the same ratio as that of x. 

5.- Test your results in the figure moving the point C to (5,2)
2.2.2. PROOF THAT THREE POINTS ARE LINEAR
In this figure we have three points P(1,4), Q(5,-2) y R(m,n)
By making appropriate changes to the point R, or changing the values of m and/or n, you can ensure that the points P, Q and R are on the same blue line, that is, LINEAR.

1.- Move the point R such that m=6, and it is aligned with P and Q. Make a note of the value of n obtained. 

2.- Copy the following calculations into your exercise book. They are the ones needed to find the value of n observed in part 1 previously: 

PQ=(5-1,-2-4)=(4,-6) 
QR=(6-5,n+2)=(1,n+2) 
 
   n+2= -6/4 ; n= -3.5

3.- Now move the point R such that n=6, and it is aligned with P and Q. Make a note of the value of m obtained. 

4.- Write down in your exercise book the calculations needed to obtain the value of m observed in part 3 previously. 

5.- In the figure move the point R to a position which will make P, Q y R linear, and then make a note of the coordinates of R observed, prove, by calculation, that the coordinates of the vectors PQ and QR are proportional.
3. MIDPOINT OF A SEGMENT
In this figure the sum of the vectors: OA + OB = OS appears. OS being the diagonal of the parallelogram OASB. The diagonals intersect eachother at their mid points. Therefore: , where A=(x1,y1) and B(x2,y2).

The coordinates of the mid point, M, of a segment between A=(x1,y1), B(x2,y2) are:

1.- Move the points A and/or B with the mouse and you can verify the coordinates of the mid point M, of the segment AB in each case.

2.-In your exercise book calculate the coordinates of the mid point of the segment between A(-3,7), B(7,-1). Verify your results using the previous figure.

3.-Calculate the symmetrical point, P', of P(8,4) with respect to Q(4,1)

Help: Q will be the mid point of the segment PP'

 Test your the result in the previous figure.
 
       
           
  Ángela Núñez Castaín
 
Ministry of Education, Social Afairs and Sport. Year 2001
 
 

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