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 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. Show What Is a Centroid Formula?The centroid of a triangle is the center of the triangle. It is referred to as the point of concurrency of medians of a triangle. Centroid FormulaThe centroid formula of a given triangle can be expressed as, C = \( \left(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\right)\) where,
Derivation of Centroid FormulaWe 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 represent the centroid of a triangle as G. Since, D is the midpoint of side PQ, applying the midpoint formula, we get its coordinates as, 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 G as, X-coordinate of G: [(2(\(x_1\) + \(x_2\))/2) + 1(\(x_3\))]/(2+1) = (\(x_1\) + \(x_2\) + \(x_3\))/3 Y-coordinate of G: [(2(\(y_1\) + \(y_2\))/2) + 1(\(y_3\))]/(2+1) = (\(y_1\) + \(y_2\) + \(y_3\))/3 Therefore, the coordinates of G are given as, ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/3)
Want to find complex math solutions within seconds? Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps. Book a Free Trial Class Let us have a look at a few solved examples to understand the centroid formula better. Examples Using Centroid FormulaExample 1: Vertices of the triangle are (4,3), (6,5), and (5,4). 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 = \(\left(\dfrac{x_1 + x_2 + x_3}{3} , \dfrac{y_1 + y_2 + y_3}{3}\right)\) = \( \left(\dfrac{4 + 6 + 5}{3} , \dfrac{3 + 5 + 4}{3} \right)\) = \(\dfrac{15}{3} , \dfrac{12}{3}\) = (5, 4) Answer: The centroid of a triangle is (5, 4). Example 2: If the coordinates of the centroid of a triangle are (3, 3) and the vertices of the triangle are (1, 5), (-1, 1), and (k, 3), then find the value of k. Solution: To find: The value of k Given parameters are, The centroid of a triangle is (3, 3) \((x_1, y_1) = (1, 5)\) \((x_2, y_2) = (-1, 1)\) \((x_3, y_3) = (k, 3)\) Using the centroid formula, The centroid of a triangle = \(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\) (3, 3) = \(\dfrac{1+(-1)+ k}{3} , \dfrac{5+1+3}{3}\) (3, 3) = \(\dfrac{k}{3} , \dfrac{9}{3}\) Equating the x-coordinates, \(\dfrac{k}{3} = 3\) k = 9 Answer: The value of k is 9. Example 3: Calculate the centroid of a triangle with vertices (1,3), (2,1), and (3,2). Solution: To find: Centroid of a triangle Using the centroid formula, we know, Centroid, G = \(\dfrac{x_1+ x_2+ x_3}{3} , \dfrac{y_1+ y_2+ y_3}{3}\) ⇒ G = ((1+2+3)/3, (3+1+2)/3) = (2,2) Answer: Centroid of given triangle = G(2, 2)
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 as, 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 find the coordinates of the centroid, G by finding the coordinates of a point that would divide the median in ratio 2:1 by applying the section formula. How Can We Apply Centroid Formula to Find 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, (\(y_1\) + \(y_2\) + \(y_3\))/3), 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, G = ((\(x_1\) + \(x_2\) + \(x_3\))/3, (\(y_1\) + \(y_2\) + \(y_3\))/3). |