distance between the radii of two intersecting congruent circles with virtually no info

Question

intersecting circles

There are two intersecting congruent circles, and I only know the length of the small arcs (coloured in red the image above), is there any way to find out the distance between the radii of the two circles? Thank you. :)

Answer 1

Suppose length of the arc is l, radius of circles is r, then $r\theta = l$ where $\theta$ (in radian) is angle subtended by the arc. So $\theta = l/r$. Then the distance between center of the two circles is $2r\cos\dfrac{\theta}{2}$

Answer 2

@amitava gave you the correct answer, for the general case of any , r and . In this specific case, since is 60° [/360°=2/3/4], we can solve it using basic geometry, without trigonometric functions.

All the radii are equal, so if you connect the centers with the endpoints of the arc, you get a rhombus. In a rhombus, the diagonals bisect the, bisect each other, and are perpendicular to each other.

So if you draw the diagonals, you get a 30-60-90 right triangle where the long leg is half of the distance between the centers, and the long leg in a 30-60-90 triangle is √3/2 times the hypotenuse.

You can see a detailed step-by-step proof here:https://geometryhelp.net/distance-between-centers-overlapping-congruent-circles/