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**Hint**: The given problem can be solved by using the concept of Pythagoras theorem. As per the Pythagoras theorem the sum of the squares on the legs of the right angle triangle is equal to the square on the hypotenuse (the side opposite to the right angle of the triangle).

**:**

__Complete step-by-step answer__We can solve the given problem by representing the towns as vertices of a right-angled triangle. Where the length of the sides shows the distance between the two towns.

First draw the right-angled triangle $ABC$showing that the town B is \[13\]miles north and \[84\]miles west of town A.

The triangle looks like:

In a right-angled triangle, \[A{C^2} + C{B^2} = A{B^2}\], where \[AC\] and \[CB\] are the lengths of the legs and \[AB\] is the length of the hypotenuse. This is by the Pythagoras Theorem.

Now from the triangle

\[BC = 13\]

\[CA = 84\]

\[AB = ?\]

Where the side \[AB\]represents the distance between the two towns A and B.

By applying Pythagoras Theorem for the triangle, we get \[A{C^2} + C{B^2} = A{B^2}\]

Now substituting the above values in Pythagoras Theorem, we get

\[{13^2} + {84^2} = A{B^2}\]

On simplification we get

\[169 + 7056 = A{B^2}\]

\[A{B^2} = 7225\]

Taking square root on both the sides, we get

\[AB = 85\]

Therefore, the distance between the two towns is 85 miles.

Correct answer is option D

**So, the correct answer is “Option D”.**

**Note**: A triangle in which one angle is \[{90^{^0}}\]is called a right angle triangle or formally an orthogonal triangle. And the Pythagoras Theorem is applicable for right angled triangles only. Pythagoras Theorem can also be written as \[{c^2} = {a^2} + {b^2}\] for the following right-angle triangle.

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