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I am trying to calculate the length of a segment intersecting a circle. I have a secant which is perpendicular to a radius. I know the length of the secant and the radius. The segment I want to calculate is perpendicular to the secant, but is not the sagitta.

The segment I'm trying to calculate is shown here in red.

Segment perpendicular to secant but not sagitta:

enter image description here

I'm at a loss. How do I find the length of this segment?

Edit: As correctly pointed out, this wasn't enough info. I do also know where the unknown segment intersects the secant. So, I have the length of every segment other than the unknown one.

Thanks

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  • $\begingroup$ We need to know some extra information to establish the precise position of the red line segment since it gets shorter as it gets further away from the sagitta. $\endgroup$ – nickgard Dec 24 '17 at 13:42
  • $\begingroup$ Absolutely right. Sorry about that. I do know where the unknown segment intersects the secant. So, I have the length of ever segment other than the unknown one. I've updated the original question to reflect that. Thanks! $\endgroup$ – Rich Friedeman Dec 24 '17 at 17:28
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Think of the red segment as part of another secant perpendicular to the first secant, as in the diagram below:

diagram of circle with perpendicular secant lines

You know the radius of the circle ($R$ in the figure), the length of the secant ($2L$ in the figure), and the distance from the center of the secant to the point where the red segment intersects the secant ($m$ in the figure).

Knowing $R$ and $L,$ you can use the Pythagorean Theorem on the right triangle in the left half of the circle to find $h,$ the distance from the center of the circle to your secant line.

Notice that the thing that looks like a rectangle in the upper right quadrant of the circle actually is a rectangle: its edges are formed by two perpendicular secant lines above and on the right, and on the left and below by the radial lines perpendicular to those secant lines. So the length of the right edge of that rectangle is $h,$ just like the left edge that you computed in the previous step. Also, the bottom edge has the same length as the top edge, $m.$

Knowing $R$ and $m,$ you can use the Pythagorean Theorem on the right triangle in the bottom half of the circle to find $k.$

But the segment labeled $k$ is only the bottom half of the secant in the right half of the circle. The top half of that secant, which is also of length $k,$ is made up of one edge of the rectangle (of length $h$) and the red segment (of unknown length $x$). Therefore $k = h + x.$ Solve for $x.$

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  • $\begingroup$ This makes perfect sense. Thank you! $\endgroup$ – Rich Friedeman Dec 24 '17 at 18:53
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Let the radius of the circle be "R" and the distance to the secant from the center of the circle be "at" where $0\le a\le 1$. Set up a coordinate system where the origin is at the center of the circle and the secant line is perpendicular to the y-axis. The circle has equation $x^2+ y^2= R^2$ and the secant line $y= aR$. If the unknown line intersects that secant at $(b, aR)$ then it is a vertical line with equation $x= b$.

The point where that vertical line intersects the circle satisfies $b^2+ (y- aR)^2= b^2+ y^2- 2aRy+ aR^2= R^2$. That is equivalent to the quadratic equation $y^2- 2aRy- (1- a)R^2= 0$ which has solution, by the quadratic formula, $y= aR+ R\sqrt{4a+4(1-a)R^2}$. The length of that segment is, of course, the difference $R\sqrt{4a+4(1-a)R^2}$,

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