Question

Find the coordinates of centroid of the triangles if points D(–7, 6), E(8, 5) and F(2, –2) are the mid points of the sides of that triangle.

Answer 1

Suppose A (x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are the vertices of the triangle.

D(–7, 6), E(8, 5), and F(2, –2) are the midpoints of sides BC, AC, and AB respectively.

Let G be the centroid of ∆ABC.

D is the midpoint of seg BC.

By the mid-point formula,

Co-ordinates of D = `((x_2 + x_3)/2, (y_2 + y_3)/2)`

∴ (-7, 6) = `((x_2 + x_3)/2, (y_2 + y_3)/2)`

∴ `(x_2 + x_3)/2 = –7  "and"   (y_2 + y_3)/2 = 6`

∴ x₂ + x₃ = –14           ...(i)    and  ∴ y₂ + y₃ = 12           ...(ii)

E is the midpoint of seg AC. 

By the mid-point formula,

Co-ordinates of E = `((x_1 + x_3)/2, (y_1 + y_3)/2)`

∴ (8, 5) = `((x_1 + x_3)/2, (y_1 + y_3)/2)`

∴ `(x_1 + x_3)/2 = 8  "and"    (y_1 + y_3)/2 = 5`

∴ x₁ + x₃ = 16           ...(iii)    and  ∴ y₁ + y₃ = 10           ...(iv)

F is the midpoint of seg AB. 

By the mid-point formula,

Co-ordinates of F = `((x_1 + x_2)/2, (y_1 + y_2)/2)`

∴ (2, -2) = `((x_1 + x_2)/2, (y_1 + y_2)/2)`

∴ `(x_1 + x_2)/2 = 2  "and"    (y_1 + y_2)/2 = -2`

∴ x₁ + x₂ = 4           ...(v)    and  ∴ y₁ + y₂ = -4           ...(vi)

Adding (i), (iii), and (v),

x₂ + x₃ + x₁ + x₃ + x₁ + x₂ = –14 + 16 + 4

∴ 2x₁ + 2x₂ + 2x₃ = 6

∴ x₁ + x₂ + x₃ = 3         ...(vii)

Adding (ii), (iv), and (vi),

y₂ + y₃ + y₁ + y₃ + y₁ + y₂ = 12 + 10 – 4

∴ 2y₁ + 2y₂ + 2y₃ = 18

∴ y₁ + y₂ + y₃ = 9   ...(viii)

G is the centroid of ∆ABC.

By centroid formula,

`"Co-ordinates of G" = ((x_1 + x_2 + x_3)/3, (y_1 + y_ 2 + y_3)/3)`

`"Co-ordinates of G" = (3/3, 9/3)`      ...[From (vii) and (viii)]

Co-ordinates of G = (1, 3)

∴ The co-ordinates of the centroid of the triangle are (1, 3).