Cartesian Coordinates Lab Report

INTRODUCTION: The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed

RATIO FORMULA/SECTION FORMULA:
When a point divides a line we have to deal with two types of division i.e. internal division and external division. INTERNAL DIVISION: Let us consider the diagram below: Let R (x,y) be a point which divides the line segment joining the points P (x_1,y_1) and Q (x_2,y_2) internally in the ratio of m∶n such that PR∶RQ = m∶n.

Let us draw a perpendicular PN, QM and RS on OX respectively. Again, PT and RK are drawn parallel to OX so that they intersect RS and QM at T and K respectively.
We can observe that PTR and RKQ are similar triangles and so we can write as
PT∶RK = PR∶RQ …(i)
TR∶KQ = PR∶RQ …(ii)

⇒ PT/RK = PR/RQ = m/n [From (i)]
Since PT = NS = OS – ON = x-x_1,
And RK = SM = OM – OS = x_2-x

Therefore we can write,
( x-x_1)/(x_2-x) = m/n
⇒ nx-nx_1=mx_2-mx
⇒ mx+nx=mx_2+nx_1
⇒ x(m+n)=mx_2+nx_1
⇒ x= (mx_2+nx_1)/(m+n)

TR/KQ = PR/RQ = m/n [From (ii)]

Since TR = SR – ST = SR – NP = y-y_1,
And KQ = MQ – MK = MQ – SR = y_2-y

Therefore we can write,
( y-y_1)/(y_2-y) = m/n
⇒ ny-ny_1=my_2-my
⇒ my+ny=my_2+ny_1
⇒ y(m+n)=my_2+ny_1
⇒ y=

m : n = 1 : 1 then the coordinates of R will be
((x_1+x_2)/2,(y_1+y_2)/2)

Example 1: The coordinates of P are (3, – 2) and the coordinates of Q are (– 3, – 2). R divides PQ in the ratio of 4 : 3. Find the coordinates of R and the length of PR.
Solution: Let the coordinates of R be (x, y)
Here (x_1,y_1)= (3, – 2), (x_2,y_2)= (– 3, – 2) and m : n = 4 : 3
Now, x = (4(-3)+3(3))/(4+3)= (-12+9)/7= 3/7 y = (4(-2)+3(-2))/(4+3)= (-8-6)/7= (-14)/7= – 2
The required coordinates of Point R that divides PQ in the ratio of 4: 3 = (3/7 , – 2)

Again, PR2 = ( 3/7 – 3)2 + [– 2– (– 2)]2 = (–18/7)2 + (0)2 = 324/49 + 0 = 324/49
PR = √(324/49) = 18/7= 2.571

Example 2: If the line segment joining the points A (1, – 2) and B (– 3, 4) gets trisected at the points P1 and P2, find the coordinates of P1 and