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 \(3(300)-4(300)-5(300)\) )

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