The vertices of a triangle are (2,a), (1,b) & (c²,-3).

Prove that its centroid can't lie on the y-axis.

Find the condition that the centroid may lie on the x-axis.

Question

The vertices of a triangle are (2,a), (1,b) &
(c2,-3)

Prove that its centroid can't lie on the y-axis
Find the condition that the centroid may lie on the
x-axis

Lalit Mehra · Accepted Answer

$Coordinates of centroid of a triangle having vertices (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}) .$

Let A(2, a), B(1, b) and C(c², – 3) be the vertices of ΔABC.

∴ Centroid of ΔABC

$= (\frac{2 + 1 + c^{2}}{3}, \frac{a + b - 3}{3}) = (\frac{3 + c^{2}}{3}, \frac{a + b - 3}{3})$

1.

Any point lies on the y-axis, if its abscissa is zero.

Now, 3 + c² ≠ 0 ∀ n∈ R

$∴ \frac{3 + c^{2}}{3} \neq 0 \forall c \in R$

Since, x-coordinates of the centroid can not be zero for any real value of c, therefore, centroid of triangle can not lie on the y -axis.

2.

Any point lies on the x-axis, if its ordinate is zero.

The centroid may lies on the x-axis if,

$\frac{a + b - 3}{3} = 0 ∴ a + b - 3 = 0 \Rightarrow a + b = 3$

The vertices of a triangle are (2,a), (1,b) & (c2,-3). Prove that its centroid can't lie on the y-axis. Find the condition that the centroid may lie on the x-axis.

The vertices of a triangle are (2,a), (1,b) & (c²,-3).

Prove that its centroid can't lie on the y-axis.

Find the condition that the centroid may lie on the x-axis.