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Centroid of a Triangle - Formula, Properties and How to Find centroid of a triangle

 

In mathematics, the centroid is the centre of a plane figure. It is the mean position and equidistant from all the points present on the boundary of the given figure.

Centroid of a triangle

The centroid of a triangle is the midpoint of the triangle. It is also the centre point of the triangle which is equidistant from all the points present on the triangle. The centroid of a triangle is obtained by the intersection of its medians. The point of intersection of the three medians of a triangle is called the centroid of that triangle.

A median is any line that joins the vertex of the angles of a triangle with the midpoint of the opposite side.

The median, which joins the centroid of a triangle from the midpoint of one side of the triangle to one of the vertices of the triangle, is divided by the centroid in the ratio of 2:1. For example,

In the given figure, ABC is a triangle whose sides are AB, BC and CA. P, Q and R are points on the sides BC, AB and CA respectively. Therefore, AP, CQ and BR are medians. The point of intersection of medians AP, CQ and BR is called the centroid V of the triangle.

Then point V divides CQ in 2:1 ratio as CV:VQ = 2:1

Then point V divides BR in 2:1 ratio as BV:VR = 2:1

Then point V divides AP in 2:1 ratio as AV:VP = 2:1

Properties of the centroid of a triangle

The centroid of a triangle possesses the following properties:

  1. The centroid is the geometric centre of a triangle.
  2. It is equidistant from all the points present on the triangle.
  3. The centroid is always in the interior of a triangle.
  4. The centroid divides the median in the ratio of 2:1.
  5. The centroid of a triangle is obtained by the intersection of the medians of the triangle.

The formula for centroid of a triangle

The centroid of a triangle can be calculated by a formula. However, the formula for the centroid of a triangle is applicable only for those triangles whose vertices are given. To obtain the centroid, the coordinates of the vertices of the triangle must be known. The following formula can be used to find the centroid when the coordinates of its vertices are given

C(x,y) = (x1 + x2 + x3)/3, (y1 + y2 + y3)/3

Where, x₁, x₂ and x₃ are the x coordinates of the vertices and y₁, y, and y, are the y-coordinates of the vertices of a triangle. For example,

C(x,y) = (1+2+3)/3, (1+2+3)/3

C(x,y) = (1,1)

are the coordinates of centroid C of the triangle.

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