Coordinate Geometry For Class 10

Coordinate Geometry Class 10 Notes

In classical Mathematics, coordinate geometry is also known as  Analytic geometry or Cartesian geometry. It can be said that the study of geometry using 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.

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 be helpful in manipulating 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 in a numerical way. 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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The figure above there are two scales:

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

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

The concept of 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 20 on the x-axis and value 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:

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

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

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

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

Therefore, the distance between A(2,3) and B (4,1) is given by

\[D = \sqrt {{{\left( {2 - 4} \right)}^2} + {{\left( {3 - 1} \right)}^2}} \]

\[{\text{   }} = \sqrt {{{\left( { - 2} \right)}^2} + {{\left( 2 \right)}^2}} \]

\[{\text{   }} = \sqrt {4 + 4}  = \sqrt 8  = 2\sqrt 2 \]  


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:

\[x = \frac{{m{x_2} + n{x_1}}}{{m + n}},y = \frac{{m{y_2} + n{y_1}}}{{m + n}}\]

 

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Mid-Point 

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

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

\[x = \frac{{{x_2} + {x_1}}}{2},y = \frac{{{y_2} + {y_1}}}{2}\]

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 the 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

\[A = \frac{1}{2}\left[ {{x_1}\left( {{y_2} - {y_3}} \right){x_2}\left( {{y_3} - {y_1}} \right) + {x_3}\left( {{y_1} - {y_2}} \right)} \right]\]

 

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

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Here is how to Solve the line segment bisection, trisection, and four-section problem.

  1. You will be given the coordinates of the two-point A and  B

  2. To find bisection, you can simply find the midpoint using its formula.

  3. 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 point P and Q

  1. 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 point P, Q and R.

FAQ (Frequently Asked Questions)

Question 1: Who is Known as the Father of Coordinate Geometry?

Answer:  René Descartes was called the father of coordinate geometry. He was also known as the father of analytical geometry due to his contributions to the field.

Question 2: What do We Need to Coordinate Geometry For?

Answer: We need to Coordinate geometry to offer a connection between algebra and geometry by using graphs of lines and curves. It is an essential branch of math and usually helps us in locating points in a plane. It also has many uses in fields of trigonometry, calculus, dimensional geometry and more.