Basics of coordinate geometry
René Déscartes, the great French mathematician of the seventeenth century, liked to lie in bed and think! One day, when resting in bed, he solved the problem of describing the position of a point in a plane. His method was a development of the older idea of latitude and longitude. In honour of Déscartes, the system used for describing the position of a point in a plane is also known as the Cartesian system.
~source ncert
Introduction
We all have learned about number lines, how to locate a point on number line and describe the position of a point on the number line.
Now, there exist many such situations in which describing the position of a point with just a single point is not possible.
Say, in your classroom you want to locate a particular bench. In this situation, it is not possible to represent a path which can properly describe the position to a bench, on a single number line.
And this is the problem which is solved by the cartesian system.
The cartesian system
To solve the above stated problem, Déscartes proposed that instead of a single number line it would require two such lines which intersects each other at their zeros and are perpendicular to each other.
And thus, we got the above figure called cartesian plane. The horizontal line is called x-axis and the vertical line is called the y-axis.
The centre or the point of intersection is called the Origin. The origin is the point of reference from where other position are described. Which means that the position of every point is described taking origin as the neutral point of reference.
Now let’s try to solve our classroom benches problem
let’s put the benches such that the bottom left bench is on the position +1 on x-axis and +1 on y-axis.
Now, we can find or describe the position of any bench with the help of our axes.
Say we want to find the position of bench A, we can see that if we go 3 unit to the positive y-axis and 2 units to the positive x-axis we will be at out bench as per this cartesian plane.
And that is indeed the position of bench A. Position of points in cartesian planes are written in round brackets as a pair of two numbers, first is call abscissa or the x-coordinate and the second is call ordinate or the y-coordinate. for example, the position of bench A will be written as A(2,3).
The cartesian system didn’t only solved the problem of describing points but also gave rise to a new form of understanding mathematics geometrically, a beautiful way to see algebraic equations in the form of graphs and shapes which is called analytical geometry.
Understanding Lines
Line is one of the simplest geometric figure. It can be defined as a one-dimensional figure that extends infinitely on both directions.
Algebraically, a line is represented by a single-variable, single-degree linear equation.
y=x
y=2x+1
y=x+1
all are the equations of a line.
Properties of a line
Slope: A line in a coordinate plane forms two angles with the x-axis, which are supplementary. Say, the angle that x-axis makes with the line in the anti-clockwise direction is θ, this angle is called the angle of inclination of the line.
If, the angle of inclination is 0°, the line is parallel to x-axis and if it is 90°, the line is perpendicular to x-axis or parallel to y-axis.
Now, if the angle of inclination is θ then the slope or gradient of a line is defined as tanθ.
The slope of a line whose inclination is 90° is not defined.
Conclusion
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Next Steps
Let’s continue our learning : Basic vector algebra
