Percy's school is due west of his house and due south of his friend Ryan's house. The distance between the school and Ryan's house is 3 kilometers and the straight-line distance between Percy's house and Ryan's house is 5.3 kilometers. How far is Percy's house from school? If necessary, round to the nearest tenth.

Answer 1

Let 𝑎 be the distance between Ryan's house and the school, and 𝑏 be the distance between Percy's house and the school.
We can form a right triangle by connecting Percy's house, the school, and Ryan's house.
Using the Pythagorean theorem, we have 𝑏^2 + 𝑎^2 = 5.3^2 (1)
Also, 𝑎^2 − 𝑏^2 = 3^2 (2)
Adding equations (1) and (2), we get
𝑏^2 + 𝑎^2 + 𝑎^2 − 𝑏^2 = 5.3^2 + 3^2
2𝑎^2 = 28.09 + 9
2𝑎^2 = 37.09
𝑎^2 = 37.09 / 2
𝑎^2 ≈ 18.545
𝑎 ≈ √18.545 ≈ 4.3 (rounded to the nearest tenth)
Substituting 𝑎 = 4.3 back into equation (2), we get
4.3^2 − 𝑏^2 = 9
18.49 − 𝑏^2 = 9
𝑏^2 = 18.49 − 9 = 9.49
𝑏 ≈ √9.49 ≈ 3.1 (rounded to the nearest tenth)
Therefore, Percy's house is about 3.1 kilometers away from school. Answer: \boxed{3.1}.