Maths Formulas Chapter-07 | Coordinate Geometry

Coordinate Geometry Class X Chapter 7

Basic formulas and complete explanation of coordinate geometry of 10th standard. It includes distance formula, section formula, mid-point formula, area of triangle area of quadrilateral and centroid of triangle

Introduction 

It is that branch of mathematics in which we solve the geometrical problems algebraically.

In coordinate geometry we combine geometry and algebra.

To locate a point in the plane we use a system of two lines which are perpendicular to each other. This system is called the cartesian coordinate system  

Cartesian coordinate system  In this system there are two lines which intersect each other at a point perpendicularly. Point of intersection of two lines is called Origin. Horizontal line is called x-axis and Vertical line is called y-axis. Numbers are written on both the axis on either side of the origin. On x-axis positive numbers are written on the right side of the origin and negative numbers are written on the left side of the origin. On y-axis positive numbers are written above the origin and negative numbers are written below the origin.

  

Coordinates of a point 

Coordinates of any point in the Cartesian plane can be written as (x, y) or (a, b). First coordinate is the value on the x-axis (or abscissa) and  second is the value on the y-axis (or ordinate). Coordinates of the origin is always (0, 0) Coordinates of any point on the x-axis is of the form (x, 0) or (a, 0) Coordinates of any point on the y-axis is of the form (0, y) or (0, b)

 SPECIFIC TOPIC FOR 10TH STANDARD ONLY

In this blog we will discuss the following point

1) Distance between two points. 2) Different types of quadrilateral. 3) Section formula. 4) Mid- point formula. 5) Area of a triangle. 6) Collinearity of three points. 7) Centroid of a triangle. 8) Area of quadrilateral.

 Distance between two points or Distance formula 

Let us suppose that two points   P(x₁, y₁) and Q(x₂, y₂) in the cartesian plane. Then distance between P and Q is denoted by |PQ| and is given by

  

or 

 

 

 Collinearity of three points by using distance formula 

Three points A, B and C are collinear if all these points lie on the straight line.

In order to prove that A, B, C are collinear we shall prove that 

|AB| + |BC| = |AC|

Types of Quadrilaterals By using Distance formula

Let ABCD is a quadrilateral. Then with the help of distance formula find the length of sides |AB|, |BC|, |CD|, |DA| and diagonals |AC|  and |BD| i) If opposite sides are equal and diagonals are not equal then quadrilateral is a parallelogram. ii) If opposite sides are equal and diagonals are also equal then quadrilateral is a Rectangle. iii) If all sides are equal and diagonals are not equal then quadrilateral is rhombus. iv) If all sides are equal and diagonals are also equal then quadrilateral is square. v) If two pair of adjacent sides are equal and diagonals are not equal then quadrilateral is a Kite.

Section Formula [Used in NCERT Ex-7.2]

It is the formula to find the coordinates of the point which divide the given line in the given ratio.

Let us suppose that two points A(x₁, y₁) and B(x₂, y₂) on a line. Let point P(x, y) divide line segment AB in the ratio   m₁ : m₂      

Therefore by section formula the coordinates of point P(x, y) is given by

 

Mid-Point Formula

When ratio become equal (or m₁ = m₂ ) then the point  P(x, y) is called the mid-point of the line segment AB and is given by

Area of Triangle and Quadrilateral [Used in NCERT Ex- 7.3]

[DELETED] 

Let A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) are the three vertices of a triangle then area of triangle is given as follows

\[Area of\Delta =\frac{1}{2}\times | x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1}+x_{2}(y_{1}-y_{2}))|\]

Another Method

Place all the points of the triangle as shown in the figure and then follow the method as shown below\[Area of\Delta =\frac{1}{2}\times |x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-(y_{1}x_{2}+y_{2}x_{3}+y_{3}x_{1}) |\]Area of triangle always be taken positive. If your answer is negative then also convert it into positive.

Collinearity of three points

Three points are collinear if area of triangle  equal to zero.

Centroid of a triangle

The point of concurrence of all the medians of a triangle is called its centroid.

If point P(x, y) is the centroid of triangle ABC then \[P(x,y)=\left ( \frac{x_{1}+x_{2}+x_{3}}{3},\; \frac{y_{1}+y_{2}+y_{3}}{3} \right )\]

Median of triangle : 

A line segment joining the vertex of the triangle with the mid-point of the opposite side is called median

Centroid divide the median in 2:1 and 2 always point towards the vertex of the triangle.

Area of quadrilateral

Place all the points of the quadrilateral as shown in the figure above and then follow the method as shown below 

Area of quadrilateral \[=\frac{1}{2}\times |x_{1}y_{2}+ x_{2}y_{3}+ x_{3}y_{4}+ x_{4}y_{1}-(y_{1}x_{2}+y_{2}x_{3}+y_{3}x_{4}+y_{4}x_{1}) |\]Area of a quadrilateral is  always be taken positive. If your answer is negative then also convert it into positive.

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