Peter lives 12 miles west of school and Bill lives north of the school

In such a question we can quickly draw the rough diagram and figure out that it is related to the right-angled triangle.

......B

P.....S

We have PS = 12,
Let, BP = x,
then, BP = BS + PS - 6. (Peter finds that the direct distance from his house (BP) to Bill's is 6 miles shorter than the distance by way of school (BS + SP))
Or, x = BS + 12 - 6
Or, BS = x - 6.

And then we can use the Pythagoras theorem.

\((x - 6)^2 + 12^2 = x^2 \)
Or, \(x^2 - 12x + 36 + 144 = x^2\)
Or, 12x = 180
Or, x = 15

Thus BS = 15 - 6 = 9.

Just to validate we can use the 9, 12, and 15 and see that they follow the Pythagorean triplet.

Note Generally the sides used for such questions follow Pythagorean triplets so that could be an easy way out to check with common Pythagorean triplet ratios such as 3:4:5 or 5:12:13 in such questions.

Since Peter lives 12 miles west of the school and Bill lives north of the school, the following figure is implied:

The distance between the two houses by way of the school = (distance between Peter's house and the school) + (distance between the school and Bill's house) = sum of the two legs.
The prompt indicates that the hypotenuse (the direct distance between the two houses) must be 6 miles shorter than the sum of the two legs (the distance by way of the school).

Since all of the values in the prompt and three of the answer choices are integers, it is almost guaranteed that the triangle will be a multiple of a 3-4-5 triangle or a 5-12-13 triangle, as follows:
3:4:5 = 6:8:10 = 9:12:15
5:12:13 = 10:24 16 = 15:36:39

Only the option in green yields a hypotenuse (15) that is 6 miles shorter than the sum of the two legs (9+12 = 21).
The following figure is implied:
Thus, the distance between Bill's house and the school = 9 miles.