How to find the Centroid of a Triangle? Last Updated : 10 Sep, 2024 Comments Improve Suggest changes Like Article Like Report Answer: The Centroid for the triangle is calculated using the formula\left (\frac{[x1+x2+x3]}{3}, \frac{[y1+y2+y3]}{3}\right)A triangle consists of three sides and three interior angles. Centroid refers to the center of an object. Coming to the centroid of the triangle, is defined as the meeting point of all 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 = 1:1The Centroid of a 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 parallelogramCG = GD, IG //  DJ, from the intercept theorem CI = IKIK = KJ, CK:IK = 2:1Therefore 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 ProblemsProblem 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).Related Articles:Centroid of a Trapezoid FormulaTypes of Center in a Triangle Comment More infoAdvertise with us Next Article Area of a Triangle in Coordinate Geometry K kandimallaommahalakshmi Follow Improve Article Tags : Mathematics School Learning Maths MAQ Similar Reads CBSE Class 10 Maths Notes PDF: Chapter Wise Notes 2024 Math is an important subject in CBSE Class 10th Board exam. So students are advised to prepare accordingly to score well in Mathematics. Mathematics sometimes seems complex but at the same time, It is easy to score well in Math. So, We have curated the complete CBSE Class 10 Math Notes for you to pr 15+ min read Chapter 1: Real NumbersReal NumbersReal Numbers are continuous quantities that can represent a distance along a line, as Real numbers include both rational and irrational numbers. Rational numbers occupy the points at some finite distance and irrational numbers fill the gap between them, making them together to complete the real line 10 min read Euclid Division LemmaEuclid's Division Lemma which is one of the fundamental theorems proposed by the ancient Greek mathematician Euclid which was used to prove various properties of integers. The Euclid's Division Lemma also serves as a base for Euclid's Division Algorithm which is used to find the GCD of any two numbe 5 min read Fundamental Theorem of ArithmeticThe Fundamental Theorem of Arithmetic is a useful method to understand the prime factorization of any number. The factorization of any composite number can be uniquely written as a multiplication of prime numbers, regardless of the order in which the prime factors appear. The figures above represent 6 min read HCF / GCD and LCM - Definition, Formula, Full Form, ExamplesThe full form of HCF/GCD is the Highest Common Factor/Greatest Common Divisor(Both terms mean the same thing), while the full form of LCM is the Least Common Multiple. HCF is the largest number that divides two or more numbers without leaving a remainder, whereas LCM is the smallest multiple that is 9 min read Irrational Numbers- Definition, Examples, Symbol, PropertiesIrrational numbers are real numbers that cannot be expressed as fractions. Irrational Numbers can not be expressed in the form of p/q, where p and q are integers and q ≠0.They are non-recurring, non-terminating, and non-repeating decimals. Irrational numbers are real numbers but are different from 12 min read Decimal Expansions of Rational NumbersReal numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. So in this article let's discuss some rational and irrational numbers an 6 min read Rational NumbersRational numbers are a fundamental concept in mathematics, defined as numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Represented in the form p/q​ (with p and q being integers), rational numbers include fractions, whole numbers, and terminating or repea 15+ min read Chapter 2: PolynomialsAlgebraic Expressions in Math: Definition, Example and EquationAlgebraic Expression is a mathematical expression that is made of numbers, and variables connected with any arithmetical operation between them. Algebraic forms are used to define unknown conditions in real life or situations that include unknown variables.An algebraic expression is made up of terms 8 min read Polynomial FormulaThe polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables. The General Formula of a Polynomial:f(x) = an​xn + an−1​xn−1 + ⋯ + a1​x + a0​Where,an​, an−1​, …, a1​, a0​ a 5 min read Types of Polynomials (Based on Terms and Degrees)Types of Polynomials: In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations. There are mainly four types of polynomials based on degree-constant polynomial (zero degree), linear polynomial ( 1st degree), quadratic polynomial (2n 9 min read Zeros of PolynomialZeros of a Polynomial are those real, imaginary, or complex values when put in the polynomial instead of a variable, the result becomes zero (as the name suggests zero as well). Polynomials are used to model some physical phenomena happening in real life, they are very useful in describing situation 13 min read Geometrical meaning of the Zeroes of a PolynomialAn algebraic identity is an equality that holds for any value of its variables. They are generally used in the factorization of polynomials or simplification of algebraic calculations. A polynomial is just a bunch of algebraic terms added together, for example, p(x) = 4x + 1 is a degree-1 polynomial 8 min read Factorization of PolynomialFactorization in mathematics refers to the process of expressing a number or an algebraic expression as a product of simpler factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, and we can express 12 as 12 = 1 × 12, 2 × 6, or 4 × 3.Similarly, factorization of polynomials involves expre 10 min read Division Algorithm for PolynomialsPolynomials are those algebraic expressions that contain variables, coefficients, and constants. For Instance, in the polynomial 8x2 + 3z - 7, in this polynomial, 8,3 are the coefficients, x and z are the variables, and 7 is the constant. Just as simple Mathematical operations are applied on numbers 5 min read Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari 14 min read Relationship between Zeroes and Coefficients of a PolynomialPolynomials are algebraic expressions with constants and variables that can be linear i.e. the highest power o the variable is one, quadratic and others. The zeros of the polynomials are the values of the variable (say x) that on substituting in the polynomial give the answer as zero. While the coef 9 min read Division Algorithm Problems and SolutionsPolynomials are made up of algebraic expressions with different degrees. Degree-one polynomials are called linear polynomials, degree-two are called quadratic and degree-three are called cubic polynomials. Zeros of these polynomials are the points where these polynomials become zero. Sometimes it ha 6 min read Chapter 3: Pair of Linear equations in two variablesLinear Equation in Two VariablesLinear Equation in Two Variables: A Linear equation is defined as an equation with the maximum degree of one only, for example, ax = b can be referred to as a linear equation, and when a Linear equation in two variables comes into the picture, it means that the entire equation has 2 variables presen 9 min read Pair of Linear Equations in Two VariablesLinear Equation in two variables are equations with only two variables and the exponent of the variable is 1. This system of equations can have a unique solution, no solution, or an infinite solution according to the given initial condition. Linear equations are used to describe a relationship betwe 11 min read Graphical Methods of Solving Pair of Linear Equations in Two VariablesA system of linear equations is just a pair of two lines that may or may not intersect. The graph of a linear equation is a line. There are various methods that can be used to solve two linear equations, for example, Substitution Method, Elimination Method, etc. An easy-to-understand and beginner-fr 8 min read Solve the Linear Equation using Substitution MethodA linear equation is an equation where the highest power of the variable is always 1. Its graph is always a straight line. A linear equation in one variable has only one unknown with a degree of 1, such as:3x + 4 = 02y = 8m + n = 54a – 3b + c = 7x/2 = 8There are mainly two methods for solving simult 10 min read Solving Linear Equations Using the Elimination MethodIf an equation is written in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero, it is said to be a linear equation in two variables. For example, 3x + y = 4 is a linear equation in two variables- x and y. The numbers th 9 min read Chapter 4: Quadratic EquationsQuadratic EquationsA Quadratic equation is a second-degree polynomial equation that can be represented as ax2 + bx + c = 0. In this equation, x is an unknown variable, a, b, and c are constants, and a is not equal to 0. The solutions of a quadratic equation are known as its roots. These roots can be found using method 12 min read Roots of Quadratic EquationThe roots of a quadratic equation are the values of x that satisfy the equation. The roots of a quadratic equation are also called zeros of a quadratic equation. A quadratic equation is generally in the form: ax2 + bx + c = 0Where:a, b, and c are constants (with a ≠0).x represents the variable.Root 13 min read Solving Quadratic EquationsA quadratic equation, typically in the form ax² + bx + c = 0, can be solved using different methods including factoring, completing the square, quadratic formula, and the graph method. While Solving Quadratic Equations we try to find a solution that represent the points where this the condition Q(x) 8 min read How to find the Discriminant of a Quadratic Equation?Algebra can be defined as the branch of mathematics that deals with the study, alteration, and analysis of various mathematical symbols. It is the study of unknown quantities, which are often depicted with the help of variables in mathematics. Algebra has a plethora of formulas and identities for th 4 min read Chapter 5: Arithmetic ProgressionsArithmetic Progressions Class 10- NCERT NotesArithmetic Progressions (AP) are fundamental sequences in mathematics where each term after the first is obtained by adding a constant difference to the previous term. Understanding APs is crucial for solving problems related to sequences and series in Class 10 Mathematics. These notes cover the ess 7 min read Sequences and SeriesA sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.For example:2, 5, 8, 11, 14, 10 min read Arithmetic Progression in MathsArithmetic Progression (AP) or Arithmetic Sequence is simply a sequence of numbers such that the difference between any two consecutive terms is constant.Some Real World Examples of APNatural Numbers: 1, 2, 3, 4, 5, . . . with a common difference 1Even Numbers: 2, 4, 6, 8, 10, . . . with a common di 3 min read Arithmetic Progression - Common difference and Nth term | Class 10 MathsArithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9....... is in a series which has a common difference (3 - 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 5 min read How to find the nth term of an Arithmetic Sequence?Answer - Use the formula: an = a1 + (n - 1)dWhere:an = nth term,a = first term,d = common difference,n = term number.Substitute the values of a, d, and n into the formula to calculate an.Steps to find the nth Term of an Arithmetic SequenceStep 1: Identify the First and Second Term: 1st and 2nd term, 3 min read Arithmetic Progression – Sum of First n Terms | Class 10 MathsArithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9……. is in a series which has a common difference (3 – 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 2, 8 min read Arithmetic MeanArithmetic Mean, commonly known as the average, is a fundamental measure of central tendency in statistics. It is defined as the ratio of all the values or observations to the total number of values or observations. Arithmetic Mean is one of the fundamental formulas used in mathematics and it is hig 12 min read Arithmetic Progression – Sum of First n Terms | Class 10 MathsArithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9……. is in a series which has a common difference (3 – 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 2, 8 min read Chapter 6: TrianglesTriangles in GeometryA triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional re 13 min read Similar TrianglesSimilar Triangles are triangles with the same shape but can have variable sizes. Similar triangles have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles are different from congruent triangles. Two congruent figures are always similar, bu 15 min read Criteria for Similarity of TrianglesThings are often referred similar when the physical structure or patterns they show have similar properties, Sometimes two objects may vary in size but because of their physical similarities, they are called similar objects. For example, a bigger Square will always be similar to a smaller square. In 9 min read Basic Proportionality Theorem (BPT) Class 10 | Proof and ExamplesBasic Proportionality Theorem: Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is the Basic Proportionality Theorem or BPT. BPT states that if a line is parallel to a side of a triangl 8 min read Pythagoras Theorem | Formula, Proof and ExamplesPythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem. It states that in a right-angled triangle, the square of the hypotenuse is equ 9 min read Chapter 7: Coordinate GeometryCoordinate GeometryCoordinate geometry is a branch of mathematics that combines algebra and geometry using a coordinate plane. It helps us represent points, lines, and shapes with numbers and equations, making it easier to analyze their positions, distances, and relationships. From plotting points to finding the short 3 min read Distance formula - Coordinate Geometry | Class 10 MathsThe distance formula is one of the important concepts in coordinate geometry which is used widely. By using the distance formula we can find the shortest distance i.e drawing a straight line between points. There are two ways to find the distance between points:Pythagorean theoremDistance formulaTab 9 min read Distance Between Two PointsDistance Between Two Points is the length of line segment that connects any two points in a coordinate plane in coordinate geometry. It can be calculated using a distance formula for 2D or 3D. It represents the shortest path between two locations in a given space.In this article, we will learn how t 6 min read Section FormulaSection Formula is a useful tool in coordinate geometry, which helps us find the coordinate of any point on a line which is dividing the line into some known ratio. Suppose a point divides a line segment into two parts which may be equal or not, with the help of the section formula we can find the c 14 min read How to find the ratio in which a point divides a line?Answer: To find the ratio in which a point divides a line we use the following formula x = \frac{m_1x_2+m_2x_1}{m_1+m_2}  y = \frac{m_1y_2+m_2y_1}{m_1+m_2}Geo means Earth and metry means measurement. Geometry is a branch of mathematics that deals with distance, shapes, sizes, relevant positions of a 4 min read How to find the Trisection Points of a Line?To find the trisection points of a line segment, you need to divide the segment into three equal parts. This involves finding the points that divide the segment into three equal lengths. In this article, we will answer "How to find the Trisection Points of a Line?" in detail including section formul 4 min read How to find the Centroid of a Triangle?Answer: The Centroid for the triangle is calculated using the formula\left (\frac{[x1+x2+x3]}{3}, \frac{[y1+y2+y3]}{3}\right)A triangle consists of three sides and three interior angles. Centroid refers to the center of an object. Coming to the centroid of the triangle, is defined as the meeting poi 4 min read Area of a Triangle in Coordinate GeometryThere are various methods to find the area of the triangle according to the parameters given, like the base and height of the triangle, coordinates of vertices, length of sides, etc. In this article, we will discuss the method of finding area of any triangle when its coordinates are given.Area of Tr 6 min read Chapter 8: Introduction to TrigonometryTrigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give 4 min read Unit Circle: Definition, Formula, Diagram and Solved ExamplesUnit Circle is a Circle whose radius is 1. The center of unit circle is at origin(0,0) on the axis. The circumference of Unit Circle is 2π units, whereas area of Unit Circle is π units2. It carries all the properties of Circle. Unit Circle has the equation x2 + y2 = 1. This Unit Circle helps in defi 7 min read Trigonometric Ratios of Some Specific AnglesTrigonometry is all about triangles or to be more precise the relationship between the angles and sides of a triangle (right-angled triangle). In this article, we will be discussing the ratio of sides of a right-angled triangle concerning its acute angle called trigonometric ratios of the angle and 6 min read Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t 10 min read Chapter 9: Some Applications of TrigonometryHeight and Distance | Applications of TrigonometryHeight is the measurement of an item in the vertical direction, whereas distance is the measurement of an object in the horizontal direction. Heights and Distances are the real-life applications of trigonometry which is useful to astronomers, navigators, architects, surveyors, etc. in solving proble 6 min read Like