I am trying to find a general relationship between a tangent line and a sector length.
The setup could be equated to a circle with its center at (0,-r) such that the topmost point of the circle is at the origin. There is a line originating at some point (0,h) which is angled downward to hit the circle at some point. This line is not guaranteed to be tangential to the circle. I want to figure out the general relationship between the length of the horizontal line (defined from (0,0) to the X-Intercept) and the Sector length (defined by the distance along the circumference from (0,0) to the point of intersection).
My gut instinct is that the relationship is a trigonometric relationship which relies on the ratio of the height ‘h’ and the radius ‘r’. It is fairly easy to find the Angle between the origin and a tangent point given h and r, as it is arcsin(r/(r+h)). Then the arc angle is the difference between that and 90. And I know that the ratio is 1:1 at the origin because the length for both is zero. I’ve played with a bunch of values and looked for some relationships but I’m struggling to make any progress here. And I can’t seem to find similar setups online.
Let the distance $GE$ be denoted by $a$. What can you tell about the angle $GAE$, in terms of $a$ and $h$? You can then use the cosine rule to determine the length $AD$, and then apply it again to determine the angle $GCB$. Can you finish it from here?