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# Coordinates

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125
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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