Question

suppose a park is located 2.4 miles east of your home. the library is 3.2 miles north of the park. what is the shortest distance between your home and the library? the shortest distance between your home and the library is \\(\square\\) miles. (type an integer or a decimal.)

Explanation:

Step1: Identify the right triangle

The home, park, and library form a right triangle. The horizontal leg (home to park) is \(2.4\) miles, and the vertical leg (park to library) is \(3.2\) miles. The shortest distance (home to library) is the hypotenuse.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), \(c = \sqrt{a^2 + b^2}\). Here, \(a = 2.4\) and \(b = 3.2\).
First, calculate \(a^2\) and \(b^2\):
\(a^2 = (2.4)^2 = 5.76\)
\(b^2 = (3.2)^2 = 10.24\)
Then, sum these values:
\(a^2 + b^2 = 5.76 + 10.24 = 16\)
Finally, take the square root:
\(c = \sqrt{16} = 4\)

Answer:

4