Three radio masts stand at A(1, -2, 3), B(4, 1, 9) and C(k, 7, p), with lengths in kilometres…

Step 1. The vectors. $\overrightarrow{AB} = (3, 3, 6)$ and $\overrightarrow{AC} = (k - 1, 9, p - 3)$.

Step 2. The multiple. The $y$-components give $9 = 3\lambda$, so $\lambda = 3$.

Step 3. The unknowns. $k - 1 = 9$, so $k = 10$. And $p - 3 = 18$, so $p = 21$.

Step 4. The ratio. $\overrightarrow{AC} = 3\overrightarrow{AB}$, so $AB$ is one third of $AC$ and $BC$ is two thirds. Then $AB : BC = 1 : 2$.