39. A line segment with endpoints at points $A(3 ; 2)$ and $B(12 ; 8)$ is divided into three equal parts. Determine the coordinates of the division points and the length of segment $\overline{A B}$.
Solution. We will use formula (1).

For point $C$ (Fig. 7) the

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-017.jpg?height=336&width=521&top_left_y=1058&top_left_x=582)

Fig. 7 ratio $\frac{A C}{C B}=\lambda=\frac{1}{2}$. Therefore, the coordinates of point $C$ are expressed as follows:

$$
x=\frac{3+\frac{1}{2} \cdot 12}{1+\frac{1}{2}}=6 ; y=\frac{2+\frac{1}{2} \cdot 8}{1+\frac{1}{2}}=4
$$

For point $D$ the ratio $\frac{A D}{B D}=\lambda=\frac{2}{1}=2$. Therefore, by formula (1)

$$
x=\frac{3+2 \cdot 12}{1+2}=9 ; \quad y=\frac{2+2 \cdot 8}{1+2}=6
$$

The coordinates of point $D$ can be found more easily if we consider that point $D$ is the midpoint of segment $\overline{C B}$, the endpoints of which are already known to us. Using formula (2), we get

$$
x=\frac{6+12}{2}=9 ; y=\frac{4+8}{2}=6
$$

Thus, segment $\overline{A B}$ is divided into three equal parts by points $C(6 ; 4)$ and $D(9 ; 6)$. The length $\quad A B=\sqrt{(12-3)^{2}+(8-2)^{2}}=$ $=\sqrt{117} \approx 10.82$.