Elizabeth is 13 miles far from the starting point.
The Pythagoras theorem states that the square of the longest side must be equal to the sum of the square of the other two sides in a right-angle triangle.
|AC|^2 = |AB|^2 + |BC|^2
Elizabeth rides a bicycle 5 miles east and then 12 miles north.
Let consider that AB = 5 miles and BC = 12 miles.
SO,
[tex]|AC|^2 = |AB|^2 + |BC|^2 \\\\|AC|^2 = |12|^2 + |5|^2 \\\\|AC|^2 = 144 + 25\\\\|AC|^2 = 169\\\\|AC| = 13[/tex]
Hence, She is 13 miles far from the starting point.
Learn more about Pythagoras theorem;
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