Radius of Intersecting Circles: Arc SPX Length (3/4)*\pi

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SUMMARY

The discussion centers on the geometric problem involving two intersecting circles of equal radius, where the length of arc SPX is defined as (3/4)*π. The key to solving for the radius of either circle lies in determining the central angle SX. Participants suggest constructing a right triangle using points S, X, and the midpoint of OS to analyze the relationship between the angles and sides of the triangle. This approach utilizes symmetry to establish that the radii SO, SX, and OX are equal, facilitating the calculation of the radius.

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Two circles with same radius intersect, so that the outermost point of either circle goes through the other's center. Arc SPX has length (3/4)*\pi (see attached image.)

What is the radius of either circle?

I think I could solve this problem if I knew the central angle SX, but I don't know how to identify when they hit (X). Maybe I have to form some special triangle into the image.
 

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Yes, draw some triangles. You want to know what the length of SX is, right? So make it the hypotenuse of a right triangle with vertices S,X, and the midpoint of OS. (Has to be a right triangle by symmetry). Then figure out what kind of special triangle it is and what its angles are, etc.
 
Join OS and OX.

SO and SX are the radii of one circle and OX is the radius of the other circle. Since the two circles have the same radii, OX = SX = SO.
So the triangle SOX is ...? and angle SOX = ...?
 

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