Question

Find the centroid of the triangle formed by the point (a,b), (b,c), (c,a) at the origin then find the value of a³ + b³ + c^3.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{for}\mathrm{a}\mathrm{triangle}\mathrm{the}\mathrm{coordinate}\mathrm{of}\mathrm{centroid}\mathrm{is}\mathrm{given}\mathrm{as}, \\ =\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3}{3},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3}{3}\right) \\ =\left(\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3},\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}\right) \\ \mathrm{But}\mathrm{it}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{centroid}\mathrm{is}\mathrm{at}\mathrm{origin}\mathrm{so}, \\ \frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{3}=0 \\ \mathrm{a}+\mathrm{b}+\mathrm{c}=0 \\ \mathrm{Now}\mathrm{we}\mathrm{know}\mathrm{that}\mathrm{if}\mathrm{a}+\mathrm{b}+\mathrm{c}=0\mathrm{then} \\ {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3\mathrm{abc} \end{gathered}$$