If A(2,-3) and B(-2,1) are two vertices of a triangle and third vertex moves on the line 2…

Question

If $A(2,-3)$ and $B(-2,1)$ are two vertices of a triangle and third vertex moves on the line $2x+3y=9$, then the locus of the centroid of the triangle is:

Accepted Answer

Let third vertex $(\alpha,\beta)$; centroid $(h,k)$:
$$h=\frac{2-2+\alpha}3=\frac{\alpha}3 \implies \alpha=3h, \qquad k=\frac{-3+1+\beta}3 \implies \beta=3k+2$$

Substituting into $2\alpha+3\beta=9$: $2(3h)+3(3k+2)=9 \implies 6h+9k+6=9 \implies 6h+9k=3 \implies 2h+3k=1$

Locus: $2x+3y=1$.

Solutions · 1