Coordinate $S(-2,-6)$, $T(18,9)$ What is $Q$ if the ratio is $2:3$

Question

Coordinate $S(-2,-6)$, $T(18,9)$ What is $Q$ if the ratio is $2:3$. So what I did so far:

1. $9-(-6)=15$,
2. $18-(-2)=20$,
3. $20/x = 2/3\Rightarrow x=30$, $30-2=28$,
4. $15/y = 2/3 \Rightarrow y=22.5$, $22.5-6=16.5$.

$Q$ is $(28, 16.5)$.

Is this work correct?

Answer 1

If $Q\in ST$ and $SQ:QT=2:3$ then: $$Q\left(\frac{3\cdot(-2)+2\cdot18}{5},\frac{3\cdot(-6)+2\cdot9}{5}\right)$$ or I used the following formula. $$Q(6,0).$$

Let $A(x_1,y_1)$, $B(x_2,y_2)$ and $C\in AB$ such that $AC:CB=m:n$.

Thus, $$C\left(\frac{nx_1+mx_2}{m+n}, \frac{ny_1+my_2}{m+n}\right).$$ It follows from the Thales theorem.

Indeed, $$m:n=AC:CB=(x_C-x_1):(x_2-x_C),$$ which gives $$m(x_2-x_C)=n(x_C-x_1)$$ or $$x_C=\frac{nx_1+mx_2}{m+n}.$$ Similarly we can get $$y_C=\frac{ny_1+my_2}{m+n}.$$