Question

derivation of Centroid formula along with one real life example​

Answer 1

Let centroid of the triangle the coordinates of whose vertices are given by A(x1, y1), B(x2, y2) and C(x3, y3) respectively. A centroid divides the median in the ratio 2:1. Hence, since ‘G’ is the median so that  AG/AD = 2/1. Since D is the midpoint of BC, coordinates of D are ((x2 + x3)/2, (y2 + y3)/2) By using the section formula, the coordinates of G are ([2(x2+x3)/2) +1× x1] / 2+1, ([2(y2+y3)/2) +1.y1] / 2+1) ∴Coordinates of the centroid G are[ (x1+x2+x3) / 3, (y1+y2+y3 ) / 3].

Ex:

Example 1:

Determine the centroid of a triangle whose vertices are (5,3), (6,1) and (7,8).

Solution

Given parameters are,

(x1, y1) = (5,3)

(x2, y2) = (6,1)

(x3, y3) = (7,8)

The centroid formula is given by

C = [(x1 + x2 + x3)/ 3, (y1 + y2 + y3)/ 3)

C = [(5 + 6 + 7) / 3, (3 + 1 + 8) / 3]

C = (18 / 3, 12 / 3)

C = (6, 4)

Example 2:

Calculate the centroid of a triangle whose vertices are (9,0), (2,8) and (1,4).

Solution

Given parameters are

(x1, y1) = (9, 0)

(x2, y2) = (2, 8)

(x3, y3) = (1, 4)

The centroid formula is given by,

C = [(x1 + x2 + x3)/ 3, (y1 + y2 + y3)/ 3]

C = [(9 + 2 + 1) / 3, (0 + 8 + 4) / 3]

C = (12 / 3, 12 / 3)

C = (4, 4)

Answer 2

Answer:

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