Centroid: Definition, Types, Theorem, Properties, Formulas

The centroid of a triangle is the point where all of a triangular plate's mass seems to act. Its 'centre of gravity,' 'centre of mass,' or barycenter are all terms for the same thing. The centroid is the location where the triangle's medians intersect, which is a remarkable fact. The centroid of a triangle is an important concept to grasp. The definition of centroid, the formula, the properties, and several sample problems will all be covered. A triangle is a three-sided, three-interior-angle delimited shape. 

Keyterms: Triangle centroid, Semi circle centroid, Centroid quarter circle, Cluster centroid, Triangle, Median, Scalene triangle, Isosceles triangle

Also, read: Congruence Of Triangles

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Definition of Centroid

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The centroid of a triangle is the place where the three medians of the triangle intersect. It's also known as the intersection point of all three medians. The median is the line that connects a side's midpoint to the opposing vertex of a triangle. The centroid of the triangle divides the median in a 2:1 ratio.

A triangle can be categorised into numerous varieties based on its sides and angles, such as -

• Scalene triangle
• Isosceles triangle
• Equilateral triangle
• Acute-angled triangle
• Obtuse-angled triangle
• Right-angled triangle

The video below explains this:

Coordinate Geometry Detailed Video Explanation:

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Theorem of Centroid

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According to the centroid theorem, the triangle's centroid is 2/3 of the distance between the vertex and the midpoint of the sides.

Suppose, ABC is a triangle having a centroid P. L, M and N are the midpoints of the sides of the triangle BC, CA and AB, respectively. Hence as per the theorem;

AP = 2/3 AL, CP = 2/3 CN and BP = 2/3 BM. 

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Properties of Centroid

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The centroid's characteristics are as follows:

• The centroid of an object should always be located within it.
• It is referred to as the object's centre.
• The centroid is the location at which all of the medians coincide.
• It also serves as the centre of gravity.

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Formula of Centroid

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Consider the triangle PQR, which has the vertices P(x1, y1),Q(x2, y2), and R(x3, y3) (x3, y3), As a result, the centroid is determined by taking the average of all three vertices and applying the formula below:

Centroid of a triangle = (x1+x2+x3)/3, (y1+y2+y3)/3

Also, read: Area of a Triangle

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Derivation of the Formula

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To find the centroid of a triangle, we use the section formula. Let PQR be any triangle with vertices P(x1, y1), Q(x2,y2), and R(x3,y3), and D, E, and F, respectively, be the midpoints of the sides PQ, QR, and PR. The centroid of a triangle is denoted by the letter G. Because D is the midpoint of side PQ, we can calculate its coordinates using the midpoint formula: D((x1 + x2)/2).

Derivation of the Formula

The triangle's centroid separates the medians in a 2:1 ratio. As a result, we may calculate the coordinates of G using the coordinates of D.

G's X-coordinate- [(2(x1x1 + x2x2)/2) + 1(x3x3)]/(2+1) = (x1x1 + x2x2 + x3x3)/3

G's Y-coordinate- [(2(y1y1 + y2y2)/2) + 1(y3y3)]/(2+1) = (y1y1 + y2y2 + y3y3)/3

As a result, ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3) are G's coordinates.

Also, read: Section Formula in Coordinate Geometry

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Things to Remember

• Centroid of a triangle = (x1+x2+x3)/3, (y1+y2+y3)/3
• Suppose, ABC is a triangle having a centroid P. L, M and N are the midpoints of the sides of the triangle BC, CA and AB, respectively. Hence as per the theorem; AP = 2/3 AL, CP = 2/3 CN and BP = 2/3 BM. 
• The centroid of the triangle divides the median in a 2:1 ratio.

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