Question

ABC is a triangle and G(4, 3) is the centroid of the triangle. If A = (1, 3), B = (4, b) and C = (a, 1), find ‘a’ and ‘b’. Find the length of side BC.

Answer 1

The coordinates of the vertices of ΔABC are A(1, 3), B(4, b) and C(a, 1).

It is known that A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are vertices of a triangle, then

The coordinates of centroid G = `((x_1 + x_2 + x_3)/3, (y_1 + y_2 + y_3)/3)`

Thus, the coordinates of the centroid of ABC are

`((1+4+a).3, (3+b+1)/3) = ((5+a)/3, (4+b)/3)`

It is given that the coordinates of the centroid are G(4, 3).

Therefore, we have

`(5+a)/3 = 4`

`5 + a = 12`

a = 7

`(4+ b)/3 = 3`

4 + b = 9

b = 5

Thus, the coordinates of B and C are (4, 5) and (7, 1) respectively.

Using distance formula, we have

`BC = sqrt((7-4)^2 + (1 - 5)^2)`

`= sqrt(9 + 16)`

`= sqrt25`

= 5 units