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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?

 Jun 19, 2022
 #1
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The sum of the x and y coordinates is 52.

 Jun 19, 2022
 #2
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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 \(\color{brown}\boxed{56}\)

 Jun 19, 2022

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