Question

The centroid of a triangle is (2,3) and two of its vertices are (5,6) and (-1,4) then the third vertex of the triangle is⁉

Answer 1

A(5,6)  ,  B(-1,4)  C(x,y)   G (2,3) {centroid}

For centroid :  2 = 5 + (-1) + x /3 ⇒ 6 = 4 +x ⇒ x = 2

and 3 = 6+4 +y / 3  ⇒ 9 = 10 + y ⇒ y = -1

Hence third vertex has coordinates  (2,-1).

Answer 2

The third vertex of the triangle is (2,-1).

• The centroid of the triangle is given by the formula $(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$ , where (x₁,y₁) , (x₂,y₂) , (x₃,y₃) are the vertices of the triangle.

• Given vertices are (5,6)(-1,4) and the centroid of the triangle is (2,3), we have to find the third vertex of the triangle.
• Let us assume the third vertex to be (x₃,y₃).
• Now,

• Substituting in the centroid formula to evaluate the third vertex of the triangle.

(x,y) = $(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

(2,3) = $(\frac{5-1+x_3}{3},\frac{6+4+y_3}{3})$

• Comparing both sides we evaluate x₃ and y₃,

$2$ = $\frac{5-1+x_3}{3}$

6 = 4+x₃

x₃ = 2

And

$3$ = $\frac{6+4+y_3}{3}$

9 = 10 + y₃

y₃ = -1

• Therefore the coordinates of the third vertex is (2,-1)