There are four points that are 5 units from the line y=13 and 13 units from the point (1,18). What is the sum of the x and y and coordinates of all four of these points?

Guest Jun 19, 2022

#2**0 **

Note that 2 points will have a y-coordinate of 18, and 2 will have a y-coordinate of 8.

If the y-coordinate is 18, the points are \((1 \pm 13, 18)\), which are \((-12, 18) \) and \((14,18)\)

If the y-coordinate is 8, we have a right triangle, where the hypotenuse is 13 units and 1 leg is 10, meaning the other leg is \(\sqrt{69}\).

This means the 2 points here are: \((1 \pm \sqrt{69}, 8)\), which are \((1 + \sqrt {69}, 8)\) and \((1 - \sqrt {69}, 8)\).

Adding everything up, we find the sum is \(\color{brown}\boxed{56}\)

BuilderBoi Jun 19, 2022