7. A coordinate system is superimposed on a billiard table. Gord has a yellow ball at $A(2,3)$. He is going to "bank" it off the side rail at $B(6,5)$, into the pocket at $C(2,7)$. How far will the yellow ball travel?
Answer
\(\boxed{4\sqrt{5}}\) is the total distance the yellow ball will travel.
Step 1 ：Find the distance between points A and B: \(AB = \sqrt{(6-2)^2 + (5-3)^2}\)
Step 2 ：Calculate the distance: \(AB = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)
Step 3 ：Find the distance between points B and C: \(BC = \sqrt{(2-6)^2 + (7-5)^2}\)
Step 4 ：Calculate the distance: \(BC = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)
Step 5 ：Add the distances AB and BC: \(AB + BC = 2\sqrt{5} + 2\sqrt{5} = 4\sqrt{5}\)
Step 6 ：\(\boxed{4\sqrt{5}}\) is the total distance the yellow ball will travel.
