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Centroid of a Triangle

Centroid of a triangle is the point of intersection of the lines joining the vertex of the triangle to the midpoint of the opposite sides.
 

The centroid G(x, y) of a ∆ ABC with vertices A(x1, y1), B(x2, y2) and C(x3, y3) is given by

Description: 37066.png

Description: 45399.png
 

Area of a triangle is given by

Description: 37060.png 

Description: 37054.png 

Description: 37048.png 

 

Note: If three points A, B and C are given, then

  • ABC is an equilateral triangle if AB = BC = AC.
  • ABC is an isosceles triangle if any two sides are equal.
  • ABC is a right angled triangle if the square of longest side is equal to the sum of the squares of the other 2 sides.

If 4 points A, B, C and D are given, then

  • ABCD is a square if AB = BC = CD = AD and the diagonals AC = BD.
  • ABCD is a rhombus if AB = BC = CD = AD and the diagonals are not equal.
  • ABCD is a rectangle if AB = CD and BC = AD and if the diagonals AC = BD.
  • ABCD is a parallelogram if AB = CD and BC = AD and the diagonals are not equal.

 

Example
The midpoint of the line segment is (3, 2). If (5, 6) is one end, then find the other end.
Solution
Let the other end be (xy).
Description: 37042.png 
Description: 37036.png 
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Example
If the points (0, 1), (3, –4) and (4, k) are collinear, then k = ?
Solution
We know that condition for collinearity is given by,
Description: 37024.png

 

Example
If (2, 1), (5, 7) and (–1, 4) are the vertices of a triangle, then find the centroid of the triangle.
Solution
Description: 37018.png
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Example
Find the equation of the straight line passing through the points (–5, 2) and (6, –4).
Solution
The equation of the line joining two points (x1y1) and (x2y2) is given by
Description: 37006.png 
Description: 37000.png 

 

Example
What shape do the points A(4, – 1), B(6, 0), C(7, 2) and D(5, 1) form?
Solution
Let us calculate the lengths of all sides and diagonals.
Description: 36993.png 
We can see that AB = BC = CD = AD and AC ≠ BD. Hence the given points form a rhombus.





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