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How do I find the position of the point on the circumference tagged as ?,? in terms of x and y

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up vote 1 down vote accepted

Denote the unknown point $\,(a,b)\,$ , then we have the vectors

$$A:=(x,y-2)-(x,y)=(0,-2)\,\,\,,\,\,\,B:=(a,b)-(x,y)=(a-x,b-y)$$

and now using the relation between inner product and $\cos\,$ we get:

$$\frac{\sqrt 3}{2}=\cos 30=\frac{A\cdot B}{||A||\,\,||B||}=\frac{2(y-b)}{2\cdot \sqrt{(x-a)^2+(y-b)^2}}\Longrightarrow$$

$$\Longrightarrow (x-a)^2+(y-b)^2=\frac{4}{3}(y-b)^2\Longrightarrow (x-a)^2=\frac{1}{3}(y-b)^2\Longrightarrow$$

$$\Longrightarrow x-a=\pm\frac{1}{\sqrt 3}\,(y-b)$$

I think you can take it from here (Note the double sign as you can have your unknown point to the right or to the left of the bottom one)

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@user494461 , please do note the edit to my answer, though it is a minor one. –  DonAntonio Sep 21 '12 at 12:34
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