A Centroid is the intersection point of the medians of a triangle. The centroid formula is used to determine its coordinates.
As per the revised CBSE Class 10th pattern for 2021-22, exams will now be conducted term-wise with a rationalized distribution of the Syllabus. The topic Centroid Formula is a part of Chapter 7: Coordinate Geometry of Class 10th NCERT and has a weightage of 6 marks in the Board Examination.
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What is the Centroid Formula?
Before discussing the Centroid Formula, it is important to understand some key terms-:
Let’s do it with the help of the given figure:
Centroid Formula
- Median- A line that is drawn from the vertex of a triangle to the midpoint of the opposite line.
In the given figure- AD, CE, BF are the medians where A, B, C are the vertices and D, E, F are the midpoints of BC, AB, and AC respectively.
- Centroid- The point where the three medians of a triangle intersect.
G is the centroid of the given triangle and divides the medians in the ratio 2:1.
The video below explains this:
Coordinate Geometry Detailed Video Explanation:
Derivation of Centroid Formula
As stated above, D, E, and F are the midpoints of BC, AB, and AC respectively.
The midpoint is the point that divides a line into equal halves.
Finding coordinates of D, E, and F,
Now, we know that G is the centroid and since it lies on AD, it divides it in the ratio of 2:1
AG: GD= 2:1
So, the coordinates of G are:
Therefore, the Centroid Formula is:
\(\text{Centroid Formula} = \left( \frac{(x1+x2+x3)}{3}, \frac{(y1+y2+y3)}{3} \right)\)
Centroid Formula is an easy concept and getting just the basic formulas would help you in getting good marks.
Read Also: Types of Triangles
Sample Questions
Question-1: If the coordinates of the centroid of a triangle with vertices (3,p), (2/3, 4 ), (-1,5) are (5,-2). Find p. (2 Marks)
Answer:
Using the Centroid Formula and substituting the values,
Now,
The given coordinates of the Centroid are (5, -2) and the calculated are (2/3, (p+9)/3)
Since we need to find the value of p, which is in the y coordinate, we will equate the y coordinate of both the values:
-2 = (p + 9)/3
(p + 9)/3 + 2 = 0
(p + 9 + 6)/3 = 0
p + 15 = 0
p = -15
Therefore, the value of p is -15
Read Further:
Question-2: Two vertices of a triangle having centroid at the origin are (-2, 3), (5,7). Find the coordinates of the third vertex. (2 Marks)
Answer:
Let the third vertex be (p,q)
Putting the values given values in the centroid formula,
Now we are given that the centroid is at origin which means it is (0,0)
Equating the given value to the calculated one for both x and y coordinates,
Therefore, the third vertex is (-3,10)
Question-3: The centroid of ?ABC, in which A (a, b), B(b, c), C(c, a) is at the origin, then calculate the value of (a3 + b3 + c3). (3 Marks)
Answer:
We know that the centroid is at the origin which means its coordinates are (0,0)
Also, x and y coordinates calculated using the formula are similar (a+b+c)/3
So, we can compare one of what we got from the formula with that of the origin,
Let’s take the x coordinate,
0 = (a + b + c)/3
a + b + c = 0
Now,
We know that a3 + b3 + c3 -3abc= (a+b+c) ( a2 + b2 + c2 -ab-bc-ca) andfrom what we have calculated, we know that a+b+c=0
So,
a3 + b3 + c3 -3abc= 0
a3 + b3 + c3 = 3abc
Therefore, the value of a3 + b3 + c3 is 3abc
Question-4:Two vertices of a triangle having an area of 6 square units are (4, 1) and (6, -3) and the third vertex lies on y = x+ 3. What will be the third vertex? (3 Marks)
Read Also: Minor and Cofactors
Answer: Formula Used-:
\(\text{Area of a Triangle} = \frac{1}{2} [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]\)
Putting the values in the given formula of area of a triangle,
Now we have two equations,
Eq. I: y = x+3 which was given
Eq. II: 15 = 2x-y which we have calculated
Putting the value of y from Eq. I in Eq. II,
15 = [-y + 2x] (Eq. II)
15 = [-(x + 3)] + 2x (Eq. I: y = x + 3)
15 = (-x – 3) + 2x
15 = x – 3
x = 18
Now, putting the value of x is Eq. I,
y = x + 3
y = 18 + 3
y = 21
Therefore, the third vertex is (18,21)
Question-5: A triangle ABC has three vertices with coordinates (5, 2), (-1, 4), and (8,0) respectively. Find the centroid and the length of the median through vertex A. (5 Marks)
Answer:
Given-: x1= 5 y1= 2
x2= -1 y2= 4
x3= 8 y3= 0
To find-: Centroid and Length of the median through A
Formulas Used-:
\(\text{Centroid Formula} = \left( \frac{(x1+x2+x3)}{3}, \frac{(y1+y2+y3)}{3} \right)\)
\(\text{Midpoint Formula} = \left( \frac{x1+x2}{2}, \frac{y1+y2}{2} \right)\)
\(\text{Area of a Triangle} = \frac{1}{2} [x1(y2-y3)+x2(y3-y1)+x3(y1-y2)]\)
Solution-:
Let ABC be a triangle and G be the centroid where A=(x1,y2), B=(x2,y2), C=(x3,y3)
Putting the values of (x1,y1), (x2,y2), (x3,y3) in the Centroid Formula,
Therefore, the coordinates of Centroid G= (4,2)
Now, we know that D is the midpoint of BC, Using the midpoint formula, the coordinates of D are:
Now using the distance formula to find the length of the median AD,
Therefore, the length of the median AD= 3/2
Things to Remember-:
- Centroid- The point which is common to all three medians.
- Centroid divides each median in the ratio of 2: 1
Read More: Area of Triangles
Important Formulas:-
Mathematics Related Links:
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