Question

If the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=3$ meets the coordinates axes at A, B and C, then the coordinates of the centroid of ∆ABC are ______________.

Answer 1

Given plane, $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=3$

Meets x-axis when y = 0 = z , $\frac{x}{a}=3$

i.e. x = 3a

Plane meets y-axis when x = 0 = z,

$$\begin{gathered} \mathrm{i}.\mathrm{e}.\frac{y}{b}=3 \\ \mathrm{i}.\mathrm{e}.y=3b \end{gathered}$$

and this plane meets z-axis when x = 0 = y

$$\begin{gathered} \mathrm{i}.\mathrm{e}.\frac{z}{c}=3 \\ \mathrm{i}.\mathrm{e}.z=3c \end{gathered}$$

∴ A is given by (3a, 0, 0)

B is given by (0, 3b, 0)

C is given by (0, 0, 3c)

$\therefore \mathrm{Centroid}\mathrm{is}\left(\frac{3a+0+0}{3},\frac{0+3b+0}{3},\frac{0+0+3c}{3}\right)$

i.e. Centroid of ΔABC = (a,b,c)