Centroid of a Triangle Centroid of a Triangle Understanding of Median is important before knowing Centroid . Median is the line joining the vertex with the midpoint of the opposite side. Centroid of a Triangle is Point of intersection of all its medians it is also called as Center of gravity In the triangle ABC , three median starting from the vertex of the angles and intersecting the sides of the opposite angles .they are meeting at a point , their intersection points are called as Centroid of the triangle . Medians are dissecting the opposite sides in equal parts and Centroid Divided the medians in the ratio of 2:1 and the longest part is near the vertex which shows that Centroid is located perpendicular on the each angle and opposites side that’s why it is diving the side in the 1/3 ratio Due to this property we can find the coordinates of the three vertices. So , if three vertices are x=(a1, b1), y(a2, b2), z=(a3 , b3) then centroid will be : C=1/3(x+y+z) C=1/3(a1+a2+a3), 1/3(b1+b2+b3)
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Centroid can also determine the area of triangle which is 1.5 times of the length of any side times to the perpendicular distance fro the centroid side Centroid is always inside the triangle and every median Divides the triangle into two equal triangles. Centroid is Sometimes called as median point too Understanding of the Centroid with the help of Examples: Example 1:Find the centroid of the triangle with vertices (1,3), (2,4), and (5,3)? Solu: C= (1+2+5)/3, (3+4+3)/3 C=(8/3, 10/3) Example 2:Find the centroid of the triangle with vertices (2,1), (3,5), and (4,3)? Solu: C= (2+3+4)/3, (1+5+3)/3 C=(9/3, 9/3) C=(3,3) Example3: Find the centroid of the triangle with vertices (-1,-2), (3,1), and (-5,5)? Solu: Centroid Co ordinates= (-1+3-5)/3, (-2+3+5)/3 C=(-3/3, 6/3) C=(-1,2) Example4: Find the values of one vertices of the triangle ABC where Centroid value is (2,3) , A(5, 1), B(4, 8).Find the value of C?
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Solu: Centroid= (x1+x2+x3)/3, (y1+y2+y3)/3 (2,3) =(5+4+x3)/3,(1+8+y3)/3 (2,3)=(9+x3)/3,(9+y3)/3 (9+x3)/3=2 9+x3=6 X3=-3 (9+y3)/3=3 9+y3=9 Y3=0 So C(-3, 0) Example5: Find the values of one vertices of the triangle PQR where Centroid value is (3,6) , P(1,3), Q(2,8).Find the value of R? Solu: Centroid= (x1+x2+x3)/3, (y1+y2+y3)/3 (3,6) =(1+2+x3)/3,(3+8+y3)/3 (3,6)=(3+x3)/3,(11+y3)/3 (3+x3)/3=2 3+x3=6
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