Question

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, -5, 7) and (-1, 7, - 6) respectively, find the coordinates of the point C.

Answer 1

Given: The coordinates of the A and B of the triangle ABC are (3, -5, 7) and (-1, 7, -6) respectively. The centroid of the triangle is (1, 1, 1) 

To find: the coordinates of vertex C 

Formula used: 

Centroid of triangle ABC whose vertices are A(x₁, y₁, z₁), B(x₂, y₂, z₂) and C(x₃, y₃, z₃) is given by

   \(\Big(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_2}{3}\Big)\)

Here A(3, -5, 7) and B(-1, 7, -6) 

Centroid of the triangle

⇒ (1,1,1) =  \(\Big(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_2}{3}\Big)\)

 ⇒ (1,1,1) =  \(\Big(\frac{3-1+x_3}{3},\frac{-5+7+y_3}{3},\frac{7-6+z_2}{3}\Big)\)

  ⇒ (1,1,1) =  \(\Big(\frac{2+x_3}{3},\frac{2+y_3}{3},\frac{1+z_2}{3}\Big)\) 

On comparing:

\(\frac{2+x_3}{3}\) =1 

⇒ 2 + x₃ = 3 

⇒ x₃ = 3 – 2 

⇒ x₃ = 1 

 \(\frac{2+y_3}{3}\) =1 

⇒ 2 + y₃ = 3 

⇒ y₃ = 3 – 2 

⇒ y₃ = 1 

 \(\frac{2+z_3}{3}\) =1 

⇒ 1 + z₃ = 3 

⇒ z₃ = 3 – 1 

⇒ z₃ = 2

Hence, coordinates of vertex C(1, 1, 2)