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# Plz Halp Asap

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There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points?

Apr 30, 2020

#1
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The sum of $$x$$ and $$y$$ coordinates of the four points is $$80$$.

Since the distance between $$(7,13)$$ and a point is $$13$$ units, and the distance from $$y = 13$$ is $$5$$, then it must be $$5$$ units above or below the line. So, this must take the shape of a triangle, so we can use the pythagorean theroem to determine where the points lie. Since $$13^{2} = 169$$, and $$5^{2} = 25$$$$169-25=144$$. And the square root of $$144$$ is $$12$$, then we will have formed a triangle with base length $$12$$ and height $$5$$, to create a triangle with hypotenuse length $$13$$. So, if we work it out, we are left with:

$$\left(-5,18\right) \left(-5,8\right) \left(19,18\right) \left(19,8\right)$$

And if we add them all up, we get $$80$$.

- Keep working,

A fellow AoPSer.

Jun 2, 2020