Village railway distance

The picture shows three villages, A, B, and C, and their mutual air distances. The new straight railway line is to be built so that all the villages are the same distance from the line and that this distance is the smallest possible. How far will they be from the track?
a = 9 km
b = 12 km
c = 15 km

Final Answer:

x₁ =  3.6 km
x₂ =  3.6 km
x₃ =  3.6 km

Step-by-step explanation:

$$\begin{gathered} a=9\text{km} \\ b=12\text{km} \\ c=15\text{km} \\ {c}_2=\sqrt{a^2+b^2}=\sqrt{9^2+12^2}=15\text{km} \\ \gamma =90{}^{\circ } \\ S=\frac{a\cdot b}{2}=\frac{9\cdot 12}{2}=54{\text{km}}^2 \\ S=\frac{c\cdot h}{2} \\ h=2\cdot S/c=2\cdot 54/15=\frac{36}{5}=7.2\text{km} \\ {x}_1=h/2=7.2/2=3.6\text{km} \end{gathered}$$

$x_2=h/2=7.2/2=3.6\text{km}$

$x_3=h/2=7.2/2=3.6\text{km}$

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