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If a right triangle has one side that is 500, another side that is 208, and the last side with a radius of 705, what is the formula to determine the length of the intersection point (y) at any given value of the x axis?

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What do you mean by "and the last side with a radius of 705"? is the curve shown part of a circle with radius = 705? This can't be because you show value 208 at the other end the O. So, the radius is either 208 or 705. Which one is correct? – NoChance Oct 2 '12 at 20:42

If the center of the circle is assumed vertically above the left vertex (which fits well as the difference between $(705-208)^2+500^2$ and $705^2$ is "negligibly small" $16$), then $$ y=\sqrt{705^2-250^2}-\sqrt{705^2-500^2}\approx 161.$$

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