Important Definitions & Formulas Coordinate Geometry - (Maths) Class 10

Points on a Cartesian Plane

A pair of numbers locate points on a plane called the coordinates. The distance of a point from the y-axis is known as abscissa or x-coordinate. The distance of a point from the x-axis is called ordinate or y-coordinate.

Example: Consider a point P(3, 2), where 3 is the abscissa, and 2 is the ordinate. 3 represents the distance of point P from the y-axis, and 2 represents the distance of point P from the x-axis.

Distance Formula

Distance between Two Points on the Same Coordinate Axes

The distance between two points that are on the same axis (x-axis or y-axis) is given by the difference between their ordinates if they are on the y-axis, else by the difference between their abscissa if they are on the x-axis.

Distance AB = 6 – (-2) = 8 units
Distance CD = 4 – (-8) = 12 units

Distance between Two Points Using Pythagoras Theorem

Let P(x₁, y₁) and Q(x₂, y₂) be any two points on the cartesian plane.
Draw lines parallel to the axes through P and Q to meet at T.
ΔPTQ is right-angled at T.
By Pythagoras Theorem,
PQ² = PT² + QT²
= (x₂ – x₁)² + (y₂ – y₁)²
PQ = √[x₂ – x₁)² + (y₂ – y₁)²]

Distance Formula: Distance between any two points (x₁, y₁) and (x₂, y₂) is given by

Where d is the distance between the points (x₁,y₁) and (x₂,y₂).

Section Formula

If the point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m1:m2, then, the coordinates of P are given by the section formula as:

Finding Ratio given the points

To find the ratio in which a given point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂),

• Assume that the ratio is k : 1
• Substitute the ratio in the section formula for any of the coordinates to get the value of k.When x1, x2 and x are known, k can be calculated. The same can be calculated from the y- coordinate also.
  

Example: Find the ratio when point (– 4, 6) divide the line segment joining the points A(– 6, 10) and B(3, – 8)?
Solution: Let the ratio be m:n.
We can write the ratio as:
m/n : 1 or k:1
Suppose (-4, 6) divide the line segment AB in k:1 ratio.
Now using the section formula, we have the following;

-4 = (3k-6)/(k+1)
– 4k – 4 = 3k – 6
7k =2
k:1 = 2:7
Thus, the required ratio is 2:7.

Midpoint

The midpoint of any line segment divides it in the ratio 1:1.

The coordinates of the midpoint(P) of line segment joining A(x₁, y₁) and B(x₂, y₂) is given by 


Example: What is the midpoint of line segment PQ whose coordinates are P (-3, 3) and Q (1, 4), respectively.
Solution: Given,  P (-3, -3) and Q (1, 4) are the points of line segment PQ.
Using midpoint formula, we have;

= (-2/2, 1/2)
= (-1, 1/2)

Points of Trisection

To find the points of trisection P and Q, which divides the line segment joining A(x₁, y₁) and B(x₂, y₂) into three equal parts:
i) AP : PB = 1 : 2

ii) AQ : QB = 2 : 1 


Example: Find the coordinates of the points of trisection of the line segment joining the points A(2, – 2) and B(– 7, 4).
Solution: Let P and Q divide the line segment AB into three parts.
So, P and Q are the points of trisection here.
Let P divides AB in 1:2, thus by section formula, the coordinates of P are (1, 0)
Let Q divides AB in 2:1 ratio, then by section formula, the coordinates are (-4,2)
Thus, the point of trisection for line segment AB are (1,0) and (-4,2).

Centroid of a Triangle

If A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) are the vertices of a ΔABC, then the coordinates of its centroid(P) are given by


Example: Find the coordinates of the centroid of a triangle whose vertices are given as (-1, -3), (2, 1) and (8, -4)
Solution: Given,
The coordinates of the vertices of a triangle are (-1, -3), (2, 1) and (8, -4)
The Centroid of a triangle is given by:
Therefore, the centroid of a triangle, G = (3, -2)

Area from Coordinates

Area of a Triangle given its vertices

If A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are the vertices of a Δ ABC, then its area is given by
Where A is the area of the Δ ABC.

Example: Find the area of the triangle ABC whose vertices are A(1, 2), B(4, 2) and C(3, 5).
Solution:
Using the formula given above,
Area = 9/2 square units.
Therefore, the area of a triangle ABC is 9/2 square units.
To know more about the Area of a Triangle,

Collinearity Condition

If three points A, B and C are collinear and B lies between A and C, then,

• AB + BC = AC. AB, BC, and AC can be calculated using the distance formula.
• The ratio in which B divides AC, calculated using the section formula for both the x and y coordinates separately, will be equal.
• The area of a triangle formed by three collinear points is zero.