Table of contents  
Terminologies related to Cartesian Plane  
Distance Formula  
Section Formula  
Finding Ratios given the Points  
Mid Point of a Line Segment  
Points of Trisection  
Centroid of a Triangle 
The distance between two points P(x_{1}, y_{1}) and Q (x_{2}, y_{2}) is given by
Note. If O is the origin, the distance of a point P(x, y) from the origin O(0, 0) is given by :
Example: Find the distance between the following points:
(i) (1, 2) and (2, 3)
(ii) (0, 1) and (6, –1)
Solution:
(i) Let the distance between the points (1, 2) and (2, 3) be d, then
d = √[(2 – ( –1))2 + (3 – 2)2] = √[9 + 1] = √10 units.
(ii) Let the distance between the points (0, 1) and (6, –1) be d, then
d = √[(6 – 0)2 + ( –1 – 1)2] = √[36 + 4] = √40 = 2√10 units.
The coordinates of the point which divides the line segment joining the points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) internally in the ratio m : n are:
The above formula is section formula.
The ratio m: n can also be written as m/n : 1 or k:1, The coordinates of P can also be written as P(x, y) =
Example: Find the coordinates of the point which divides the line segment joining the points (4,6) and (5,4) internally in the ratio 3:2.
Sol: Let P(x, y) be the point which divides the line segment joining A(4, 6) and B(5, 4) internally in the ratio 3 : 2.
Here,
(x1, y1) = (4, 6)
(x2, y2) = (5, 4)
m : n = 3 : 2
Using the section formula,
Coordinates of P are,
Therefore, P(x,y) = (7/5,0).
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}),
When x_{1}, x_{2} 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;
Thus, the required ratio is 2:7.
The midpoint of the line segment joining the points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is
Substituting m = 1, n = 1 in section formula we get,
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).
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:
G = ((x1+x2+x3)/3, (y1+y2+y3)/3)
G = ((1+2+8)/3, (3+14)/3)
G = (9/3, 6/3)
G = (3, 2)
Therefore, the centroid of a triangle, G = (3, 2)
116 videos420 docs77 tests

1. What is the Distance Formula in the Cartesian Plane? 
2. How do you find the Midpoint of a Line Segment? 
3. What is the Section Formula and how is it used? 
4. How can you find the Points of Trisection of a Line Segment? 
5. What is the Centroid of a Triangle and how is it calculated? 

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