Terri works due north of home and Denny works due east thus we can consider the distances from home to be legs of a right triangle, with the right angle at their home, and the distance between them will be the hypotenuse of the right triangle.

let

a = 5 mi is Terri's distance from home

b = Danny's distance from home

c = the distance between Terri and Danny

a, b are the legs and c the hypotenuse of the right triangle

c = b + 1

using the Pythagorean Theorem

a^2 + b^2 = c^2

5^2 + b^2 = (b + 1)^2

25 + b^2 = b^2 + 2b + 1

2b + 1 = 15

b = 7 mi

Danny is 7 miles away from home.