06/04/2020 (2011: National Sprint Round, Problem 7)

Q: The point (6, 2) lies on a line with slope equal to 2/5. What is the distance between (6,2) and either one of the two closest points on the line with integer coordinates? Express your answer in simplest radical form.

A: √29

We could graph the nearest points that make a 2/5 slope with (6,2) as shown below and use the distance formula to find the length between each.

We'd see that the horizontal difference is 5 and the vertical difference is 2, and add the squares of both. 5² + 2² = 25 + 4 = 29. You find the square root of it and you get √29.

OR

You could apply the fact that 2/5 slope means that the next point with integer coordinates form a triangle (with the current point) with dimensions of 2 x 5.

You'd add the square of both 2 and 5 and get 4 + 25 = 29. You'd then find the square root of 29, which is √29.