how to find centroid of a triangle

Centroid Formula

The geometric center of the object is known as the centroid. For determining the coordinates of the triangle's centroid we use the centroid formula. The centroid of a triangle tin can be determined as the point of intersection of all the three medians of a triangle. The centroid of a triangle divides all the medians in a 2:1 ratio. Let us learn about the centroid formula with few solved examples at the end.

What Is a Centroid Formula?

The centroid of a triangle is the eye of the triangle. It is referred to as the bespeak of concurrency of medians of a triangle.

Centroid Formula

The centroid formula of a given triangle tin can be expressed equally,

C = \( \left(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{iii}\right)\)

where,

C denotes the centroid of a triangle
\(x_1, x_2, x_3\) are the x-coordinates of the 3 vertices.
\(y_1, y_2, y_3\) are the y-coordinates of the three vertices.

centroid formula

Derivation of Centroid Formula

We apply the section formula to derive the centroid of a triangle formula. Let PQR be any triangle with the coordinates of vertices as P(\(x_1\), \(y_1\)), Q(\(x_2\),\(y_2\)), and R(\(x_3\),\(y_3\)), such that D, E, and F are midpoints of the side PQ, QR, and PR respectively. We correspond the centroid of a triangle as G. Since, D is the midpoint of side PQ, applying the midpoint formula, nosotros get its coordinates as,
D((\(x_1\) + \(x_2\))/2)

centroid of a triangle formula derivation

The centroid of a triangle divides the medians in the ratio 2:1. Therefore, from the coordinates of D, we can find the coordinates of M every bit,

X-coordinate of G: [(two(\(x_1\) + \(x_2\))/2) + 1(\(x_3\))]/(2+1) = (\(x_1\) + \(x_2\) + \(x_3\))/iii

Y-coordinate of Thousand: [(2(\(y_1\) + \(y_2\))/two) + ane(\(y_3\))]/(2+i) = (\(y_1\) + \(y_2\) + \(y_3\))/3

Therefore, the coordinates of K are given every bit, ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/3)

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Let united states of america have a look at a few solved examples to empathise the centroid formula ameliorate.

Examples Using Centroid Formula

Example 1:Vertices of the triangle are (iv,3), (6,5), and (5,four). Determine the centroid of a triangle using the centroid formula.

Solution:

To find: Centroid of a triangle.

Given parameters are,

\((x_1, y_1) = (4,3)\)

\((x_2, y_2) = (6,5)\)

\((x_3, y_3) = (5,4)\)

Using centroid formula,

The centroid of a triangle = \(\left(\dfrac{x_1 + x_2 + x_3}{3} , \dfrac{y_1 + y_2 + y_3}{3}\right)\)

= \( \left(\dfrac{4 + half dozen + 5}{3} , \dfrac{three + 5 + 4}{3} \right)\)

= \(\dfrac{15}{3} , \dfrac{12}{3}\)

= (5, 4)

Answer:The centroid of a triangle is (5, iv).

Example ii: If the coordinates of the centroid of a triangle are (3, iii) and the vertices of the triangle are (1, v), (-1, ane), and (one thousand, 3), then find the value of k.

Solution:

To find: The value of k

Given parameters are,

The centroid of a triangle is (three, three)

\((x_1, y_1) = (1, v)\)

\((x_2, y_2) = (-1, 1)\)

\((x_3, y_3) = (k, 3)\)

Using the centroid formula,

The centroid of a triangle = \(\dfrac{x_1+ x_2+ x_3}{three} , \dfrac{y_1+ y_2+ y_3}{3}\)

(iii, iii) = \(\dfrac{ane+(-ane)+ g}{3} , \dfrac{five+1+3}{3}\)

(3, three) = \(\dfrac{one thousand}{3} , \dfrac{ix}{3}\)

Equating the x-coordinates,

\(\dfrac{chiliad}{iii} = 3\)

k = ix

Reply: The value of k is 9.

Example iii:Summate the centroid of a triangle with vertices (1,3), (ii,1), and (3,2).

Solution:

To find: Centroid of a triangle

Using the centroid formula, we know, Centroid, Grand = \(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\)

⇒ K = ((1+2+three)/3, (3+one+two)/three) = (2,two)

Respond: Centroid of given triangle = M(2, 2)

FAQs on Centroid Formula

What Is Meant By Centroid Formula?

The centroid formula is the formula used for the calculation of the centroid of a triangle. Centroid is the geometric center of any object. The centroid of a triangle refers to that point that divides the medians in 2:1. Centroid formula is given every bit,
G = ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/3)
where, (\(x_1\), \(y_1\)), (\(x_2\), \(y_2\)), and (\(x_3\), \(y_3\)) are the coordinates of the vertices

How to Derive the Centroid of a Triangle Formula?

We can derive the centroid of a triangle formula using the section formula. We can discover the coordinates of the centroid, G by finding the coordinates of a indicate that would divide the median in ratio 2:1 by applying the section formula.

How Can We Apply Centroid Formula to Detect Centroid of a Triangle?

We can apply the section formula to find the centroid of the triangle, given the coordinates of the vertices. The formula is given as, G = ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/three), where (\(x_1\), \(y_1\)), (\(x_2\), \(y_2\)), and (\(x_3\), \(y_3\)) are the coordinates of the vertices.

What Is Centroid of a Triangle Formula Used for?

The centroid of a triangle is used for the calculation of the centroid when the vertices of the triangle are known. The centroid of a triangle with coordinates (\(x_1\), \(y_1\)), (\(x_2\), \(y_2\)), and (\(x_3\), \(y_3\)) is given as, Thou = ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/3).

Source: https://www.cuemath.com/centroid-formula/

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