A pair of numbers locate points on a plane called the coordinates. The distance of a point from the yaxis is known as abscissa or xcoordinate. The distance of a point from the xaxis is called ordinate or ycoordinate.
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 yaxis, and 2 represents the distance of point P from the xaxis.
The distance between two points that are on the same axis (xaxis or yaxis) is given by the difference between their ordinates if they are on the yaxis, else by the difference between their abscissa if they are on the xaxis.
Distance AB = 6 – (2) = 8 units
Distance CD = 4 – (8) = 12 units
Let P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) be any two points on the cartesian plane.
Draw lines parallel to the axes through P and Q to meet at T.
ΔPTQ is rightangled at T.
By Pythagoras Theorem,
PQ^{2} = PT^{2} + QT^{2}
= (x_{2} – x_{1})^{2 }+ (y_{2} – y_{1})^{2}
PQ = √[x_{2} – x_{1})^{2 }+ (y_{2} – y_{1})^{2}]
Distance Formula: Distance between any two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by
Where d is the distance between the points (x_{1},y_{1}) and (x_{2},y_{2}).
If the point P(x, y) divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) internally in the ratio m1:m2, then, the coordinates of P are given by the section formula as:
To find the ratio in which a given point P(x, y) divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}),
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 = (3k6)/(k+1)
– 4k – 4 = 3k – 6
7k =2
k:1 = 2:7
Thus, the required ratio is 2:7.
The midpoint of any line segment divides it in the ratio 1:1.
The coordinates of the midpoint(P) of line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) 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)
To find the points of trisection P and Q, which divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) 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).
If A(x_{1}, y_{1}), B(x_{2}, y_{2}), and C(x_{3}, y_{3}) 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)
If A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) 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,
If three points A, B and C are collinear and B lies between A and C, then,
116 videos420 docs77 tests

1. What is the distance formula used to calculate the distance between two points on a Cartesian plane? 
2. How do you find the midpoint of a line segment between two points? 
3. What is the section formula and how is it used to find the coordinates of a point dividing a line segment in a given ratio? 
4. What is the condition for three points to be collinear on a Cartesian plane? 
5. How can I find the centroid of a triangle given its vertices? 

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