[Maths Class Notes] on Coordinate Geometry For Class 10 Pdf for Exam

In classical Mathematics, coordinate geometry is also known as Analytic geometry or Cartesian geometry. It can be said that the study of geometry uses a coordinate system. Coordinate geometry is a branch of geometry. With the help of which the position of the points on the plane is defined with the help of an ordered pair of numbers also known as coordinates.

When we consider the position of the points on a Cartesian plane with some ordered pair of numbers then this particular point is called the coordinates. These concepts are considered to be very interesting and it is said to describe the links between geometry and algebra. This is done by using graphs that involve lines and curves. Thus, this gives us a geometrical understanding of Algebra and enables us to solve problems related to geometry. The coordinates or the concept related to geometry i.e. Cartesian or coordinate geometry are explained using formulas and their derivations.

The use of coordinate geometry is found in engineering and physics. This, in turn, is used in aviation, aeronautical science and rocketry. It is also used in mapping geographical location and distances and also used in global positioning systems or GPS. Furthermore, the use of coordinate geometry is also in mobile devices, laptops and computers where it is used to find the cursor. Thus, coordinate geometry can be described as the foundation of most of the modern areas or fields of geometry which can include discrete, differential, algebraic and computing geometry. The Cartesian planes help manipulate planar equations and also straight lines and squares in two-dimension and three-dimensions. In simple terms, coordinate geometry is concerned with the representation and definition of geometrical shapes explained numerically which means that the data related to shapes are represented numerically.

Analytic geometry has wide use in physics and engineering. It is also used in aviation, rocketry, space science, and spaceflight. Coordinate geometry is the foundation of most modern fields of geometry which include algebraic, differential, discrete and computational geometry.

Often the Cartesian coordinate system is proved to help manipulate equations for planes, straight lines, and squares, in 2D and 3D. As taught to us in our school books, analytic geometry can be explained more simply as it is concerned with defining and representing geometrical shapes numerically. It extracts numerical information from shapes’ numerical definitions and representations.

Now let us understand the concept of coordinates with the help of an example.

Given below is a representation of a coordinate plane. On the left-hand side, there are rows numbered as 1, 2, 3, 4, 5, 6, and so on and the first column of the grid is labelled as A, B, C, D, E, F, etc.

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You can also see a letter X which is located in the box D3 i.e. column D and row 3. therefore, D and 3 are the coordinates of this box.

The box can be divided into rows and columns. There are several boxes in every row and several boxes in every column. So, with both of them, you can find one single box. This box will be where the rows and the columns intersect each other.

The Coordinate Plane

In coordinate geometry, all the points have to be located on the coordinate plane. Look at the figure given below.

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In the figure above there are two scales:

1) X-axis which is running across the plane.

2) y-axis which is at the right angle to the X-axis.

The concept of the X-axis and y-axis is the same as the rows and columns that we have discussed in the first part above.

The Concept of Coordinates

The point where the x and the y-axis intersect is known as the origin. At this point, both x, as well as y, are 0.
The values on the right-hand side of the x-axis are positive while the values on the left-hand side of the x-axis are negative.
Similarly, on the y-axis, the values located above the origin are positive while the values located below the origin are negative.
While locating a point on the plane, it is to be determined by a set of two numbers. First, we have to write about its location on the x-axis followed by its location on the y-axis.

Together, a single and unique position on the plane will be determined.

So, in the figure above, point A has a value of 20 on the x-axis and a value of 15 on the y-axis. These are also the coordinates of point A.

Distance Formula

The distance formula is used to calculate the distance (d) between two points. The distance formula is derived by creating a triangle and the length of the hypotenuse is taken out using the Pythagorean theorem. The hypotenuse of the triangle is the distance between the two points.

The formula for calculating the distance: the distance between two points is calculated with the help of the formula given below:

[PQ = sqrt {{{left( {{x_2} – {x_1}} right)}^2} + {{left( {{y_2} – {y_1}} right)}^2}} ]

Example: find the distance between (2,3) and (4,1)

Solution: let the points be A(2,3) and B (4,1)

Therefore, Using the formula [PQ = sqrt {{{left( {{x_2} – {x_1}} right)}^2} + {{left( {{y_2} – {y_1}} right)}^2}} ]

Then the distance, say D will be =[D = sqrt {{{left( {2 – 4} right)}^2} + {{left( {3 – 1} right)}^2}} ]

D =[sqrt{8}] =[2sqrt{2}]

Section Formula

Section formula helps us to know the coordinates of the points that divide a given line segment into two parts.

For example: let us take P(x, y) as any point on the line segment AB, which divides AB in the ratio m: n, then the coordinates of the point P(x, y) will be:

x = mx₂ + nx₁ / m + n , y = my₂ + ny₁/ m + n

Mid-Point

The midpoint of a line segment can be defined as the point which divides a line segment into two halves.

Example: let P(x, y) be the mid-point of the line segmen
t AB, which divides AB in the ratio of 1:1. The coordinates of the point P(x, y) will be:

x = x₂ + x₁ / 2 , y = y₂ + y₁ / 2

Area of Triangle ABC

We all have studied how to calculate the area of a triangle if its base and corresponding height (altitude) are given by using the formula:

Area of a triangle = 1/ 2 × base (b) × altitude(h)

Further in Class IX, we studied Heron’s formula to find the area of a triangle. Now, if the coordinates of the vertices of a triangle are given, will you be able to find its area? Well, of course, you could find the lengths of the three sides using the distance formula and then use Heron’s formula. But would it not be tedious? And what if particularly the lengths of the sides are irrational numbers. So let us see if there is an easier way out.

The area of triangle ABC having coordinates A(x1,y1) , B(x2,y2) and C(x3,y3) is given by:

A = ½ [x₁(y₂-y₃ ) x₂(y₃– y₁) + x₃( y₁-y₂) ]

For points A, B and C to be collinear, the value of A should always be zero.

Here is How to solve the Line Segment Bisection, Trisection, and Four-Section Problem.

You will be given the coordinates of the two-point A and B.
To find bisection, you can simply find the midpoint using its formula.
To find trisection(i.e., three equal parts of the line ). Let us take the points as P and Q, then AP=PQ=QB
Where P divides the line AB into 1:2 part
Whereas Q divides the line AB into 2:1 part

Hence, we can use the section formula to get the coordinate of points P and Q

To find four equal parts. Let us take the points as P, Q, and R.

Such that AP=PQ=QR=RB

Where P divides the line AB into 1:3 part

Q divides the line AB into 1:1 part

R divides the line AB into 3:1 part

Hence, we can use the section formula to get the coordinate of points P, Q and R.

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