Introduction
In this chapter, you will learn how to find the distance between two points when their coordinates are given. You will also study how to determine the coordinates of a point that divides a line segment joining two given points in a specific ratio.
Take a look at the canvas below. It is just an empty page without any details. It would be very difficult to give someone precise directions to draw something on it. For example, how would you guide someone to draw the face of a cat? You might say something like, "Start at the left corner and draw a line toward the east," and so on.
Did you draw the line?
Since a simple empty paper has no information, we cannot do much with it. What can we add to this paper to make it more useful?
A simple insight that people found is that, if you draw horizontal and vertical lines on this paper we can create an abstraction which represents a lot of things.
We have also studied earlier that to locate the position of a point on a plane, we require a pair of coordinate axes. The distance of a point from the y-axis is called its x-coordinate, or abscissa. The distance of a point from the x-axis is called its y-coordinate, or ordinate.
The coordinates of a point on the x-axis are of the form
Here is a play for you. Draw a set of a pair of perpendicular axes on a graph paper. Now plot the following points and join them as directed:
Join the point A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) to A. Then join the points P(3.5, 7), Q (3, 6) and R(4, 6) to form a triangle. Also join the points X(5.5, 7), Y(5, 6) and Z(6, 6) to form a triangle. Now join S(4, 5), T(4.5, 4) and U(5, 5) to form a triangle. Lastly join S to the points M(0, 5) and N(0, 6) and join U to the points O(9, 5) and V(9, 6). What picture have you got?
Also, we have seen that a linear equation in two variables of the form ax + by + c = 0, (a, b are not simultaneously zero), when represented graphically, gives a
Further, we have also seen that the graph of
In fact, coordinate geometry has been developed as an algebraic tool for studying geometry of figures. It helps us to study geometry using algebra, and understand algebra with the help of geometry. Thus, it acts as a link between algebra and geometry, giving shapes to abstract equations in the x's and y's. Because of this, coordinate geometry is widely applied in various fields such as physics, engineering, navigation, seismology and art!