Final answer:
To determine the distance from Brookfield to Patrick's house, the Pythagorean theorem is used. The locations form a right triangle with sides of 6 and 8 miles. The hypotenuse, representing the distance from Brookfield to Patrick's house, is found to be 10 miles.
Step-by-step explanation:
The question asks how far Brookfield is from Patrick's house, measured in a straight line. Since Patrick's house is due west of Norwood and due south of Brookfield, the locations of Patrick's house, Norwood, and Brookfield form a right triangle. We are given that Norwood is 6 miles from Patrick's house (the south side of the triangle) and 8 miles from Brookfield (the east side of the triangle).
To find the distance between Brookfield and Patrick's house (the hypotenuse of the triangle), we will apply the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is c² = a² + b².
Let's substitute the known lengths into the formula:
c² = 6² + 8²
c² = 36 + 64
c² = 100
Now we find the square root of 100 to solve for c:
c = √100
c = 10 miles
Therefore, Brookfield is 10 miles from Patrick's house, measured in a straight line.