In the figure shown below, $C$ is on the segment with endpoints A and D. The distance between $A$ and $B$ is 2,000 km, between $A$ and $C$ is 1,600 km, between $A$ and $D$ is 2,500 km, and between $B$ and $C$ is 1,200 km. What is the distance, in kilometers, between $B$ and $D$ ?

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$\overline{CD}$ can be found by subtracting $\overline{AC}$ from $\overline{AD}$

$$\overline{CD} = 2{,}500 - 1{,}600$$ $$\overline{CD} = 900$$

Since we know the two sides of $\triangle{BCD}$, we can use the pythagorean theorem to obtain the hypotenuse.

(Note: the sides also form a pythagorean triplet $3(300)-4(300)-5(300)$ )

$$\overline{CD}^2+\overline{BC}^2 = \overline{BD}^2$$ $$900^2 + 1{,}200^2 = \overline{BD}^2$$ $$\overline{BD} = \boxed{1{,}500}$$