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How to find the Centroid of a Triangle?

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A triangle consists of three sides and three interior angles. Centroid refers to the centre of an object. Coming to the centroid of the triangle, it is defined as the meeting point of all the three medians of a triangle. The median of a triangle is defined as the line that is drawn from one side of a triangle to the midpoint of another side. So, we can say that the median is a line that is drawn from a vertex to the opposite side and divides in a 1:1 ratio. 

In order to understand better about the median consider the below figure:

A line that is drawn from the vertex “A” divides the opposite side i.e., “BC” into 2 equal parts.

Therefore,  BD:DC = 1:1

The centroid of the triangle divides the median in the ratio 2:1. To prove that centroid divides median in 2:1 ratio let’s consider a triangle and reflect it on one of the sides i.e., as shown in the below fig. where triangle ABD is the reflection of triangle ACD when reflected along with the side AD.

 , 

ACDB is a parallelogram 

The lines GD = AF and AG // FD, therefore AGDF make a parallelogram

CG = GD, IG //  DJ, from the intercept theorem CI = IK

IK = KJ, CK:IK = 2:1

Therefore the centroid here is I that divides median CK in 2:1 ratio.

In order to find the coordinates of centroid, it is simply the mean of all the coordinates of three vertices of a triangle. Let us consider (x1, y1), (x2, y2) and (x3, y3) as the three coordinates of the triangle, then the coordinates of centroid are  ([x1+x2+x3]/3, [y1+y2+y3]/3).

Centroid formula for the triangle is

\left (\frac{[x1+x2+x3]}{3}, \frac{[y1+y2+y3]}{3}\right)

Sample Problems

Problem 1. Find the centroid of the triangle whose vertices are A(2,4), B(2,6) and C(4,6),

Solution:

Given A(2,4), B(2,6) and C(4,6) as the vertices of triangle ABC.

From the centroid formula of triangle we know,

centroid = ([x1+x2+x3]/3, [y1+y2+y3]/3) 

substituting the given values we get ⇒  ([2+2+4]/3, [4+6+6]/3)

⇒ (8/3,16/3)

Hence the centroid for the given vertices is (8/3,16/3).

Problem 2. Find the centroid of the triangle whose vertices are A(9,4), B(1,6) and C(-2,0),

Solution:

Given A(9,4), B(1,6) and C(-2,0) as the vertices of triangle ABC.

From the centroid formula of triangle we know,

centroid = ([x1+x2+x3]/3, [y1+y2+y3]/3)

substituting the given values we get ⇒  ([9+1+-2]/3, [4+6+0]/3)

⇒ (8/3,10/3)

Hence the centroid for the given vertices is (8/3,10/3).

Problem 3. Find the centroid of the triangle whose vertices are P(-2,-4), Q(0,2) and R(0,0).

Solution:

Given P(-2,-4), Q(0,2) and R(0,0) as the vertices of triangle PQR.

From the centroid formula of triangle we know,

centroid = ([x1+x2+x3]/3, [y1+y2+y3]/3)

substituting the given values we get ⇒  ([-2+0+0]/3, [-4+2+0]/3)

⇒ (-2/3,-2/3)

Hence the centroid for the given vertices is (-2/3,-2/3).

Problem 4. Find the centroid of the triangle whose vertices are A(2,6), B(9,4) and C(6,15)

Solution:

Given A(2,6), B(9,4) and C(6,15) as the vertices of triangle ABC.

From the centroid formula of triangle we know,

centroid = ([x1+x2+x3]/3, [y1+y2+y3]/3)

substituting the given values we get ⇒  ([2+9+6]/3, [6+4+15]/3)

⇒ (17/3,25/3)

Hence the centroid for the given vertices is (17/3,25/3).

Problem 5. Find the centroid of the triangle whose vertices are A(20,0), B(2,0) and C(11,6)

Solution:

Given A(20,0), B(2,0) and C(11,6) as the vertices of triangle ABC.

From the centroid formula of triangle we know,

centroid = ([x1+x2+x3]/3, [y1+y2+y3]/3)

substituting the given values we get ⇒  ([20+2+11]/3, [0+0+6]/3)

⇒ (33/3,6/3)

Hence the centroid for the given vertices is (11,2).


Last Updated : 15 Mar, 2022
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