Loading [MathJax]/jax/output/HTML-CSS/jax.js

Centroid of a Triangle

The Centroid of a triangle is the point of intersection of the medians of a triangle.

To find the centroid of a triangle

Let A (x1, y1), B (x2, y2) and C (x3, y3) are  the three vertices of the ∆ABC .

Let D be the midpoint of side BC.

Since, the coordinates of B (x2, y2) and C (x3, y3), the coordinate of the point D are (x2+x32, y2+y32).

Let G(x, y) be the centroid of the triangle ABC.

Then, from the geometry, G is on the median AD and it divides AD in the ratio 2 : 1, that is AG : GD = 2 : 1.

Therefore, x = {2(x2+x3)2+1x12+1} = x1+x2+x33

y = {2(y2+y3)2+1y12+1} = y1+y2+y33

Therefore, the coordinate of the G are (x1+x2+x33, y1+y2+y33)

Hence, the centroid of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) has the coordinates (x1+x2+x33, y1+y2+y33).


Note: The centroid of a triangle divides each median in the ratio 2 : 1 (vertex to base).


Solved examples to find the centroid of a triangle:

1. Find the co-ordinates of the point of intersection of the medians of trangle ABC; given A = (-2, 3), B = (6, 7) and C = (4, 1).

Solution:

Here, (x1  = -2, y1 = 3), (x2  = 6, y2 = 7) and  (x3  = 4, y3 = 1),

Let G (x, y) be the centroid of the triangle ABC. Then,

x = x1+x2+x33 = (2)+6+43 = 83

y = y1+y2+y33 = 3+7+13 = 113

Therefore, the coordinates of the centroid G of the triangle ABC are (83, 113)

Thus, the coordinates of the point of intersection of the medians of triangle are (83, 113).


2. The three vertices of the triangle ABC are (1, -4), (-2, 2) and (4, 5) respectively. Find the centroid and the length of the median through the vertex A.

Solution:

 Here, (x1  = 1, y1 = -4), (x2  = -2, y2 = 2) and  (x3  = 4, y3 = 5),

Let G (x, y) be the centroid of the triangle ABC. Then,

x = x1+x2+x33 = 1+(2)+43 = 33 = 1

y = y1+y2+y33 = (4)+2+53 = 33 = 1

Therefore, the coordinates of the centroid G of the triangle ABC are (1, 1).

D is the middle point of the side BC of the triangle ABC.

Therefore, the coordinates of D are ((2)+42, 2+52) = (1, 72)

Therefore, the length of the median AD = (11)2+(472)2 = 152 units.


3. Two vertices of a triangle are (1, 4) and (3, 1). If the centroid of the triangle is the origin, find the third vertex.

Solution:

Let the coordinates of the third vertex are (h, k).

Therefore, the coordinates of the centroid of the triangle (1+3+h3, 4+1+k3)

According to the problem we know that the centroid of the given triangle is (0, 0)

Therefore,

1+3+h3 = 0 and 4+1+k3 = 0

⟹ h = -4 and k = -5

Therefore, the third vertex of the given triangle are (-4, -5).

 Distance and Section Formulae





10th Grade Math

From Centroid of a Triangle to HOME




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Successor and Predecessor | Successor of a Whole Number | Predecessor

    Jul 29, 25 12:59 AM

    Successor and Predecessor
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More

  2. Worksheet on Area, Perimeter and Volume | Square, Rectangle, Cube,Cubo

    Jul 28, 25 01:52 PM

    Volume of a Cuboids
    In this worksheet on area perimeter and volume you will get different types of questions on find the perimeter of a rectangle, find the perimeter of a square, find the area of a rectangle, find the ar…

    Read More

  3. Worksheet on Volume of a Cube and Cuboid |The Volume of a RectangleBox

    Jul 25, 25 03:15 AM

    Volume of a Cube and Cuboid
    We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.1. Fill in the blanks:

    Read More

  4. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 24, 25 03:46 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  5. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 23, 25 11:37 AM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More