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I have a semi-circumference. There are three chords within it, one after the other. The chords measure 3, 2 and 1. Find the radius of the semi-circumference. Thanks.

Edit by DS:

enter image description here

Donald Splutterwit
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Jose B. Caceres
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  • as stated, the radius can be whichever $3 \le r$ . What do you mean by "..one after the other"? – G Cab Aug 18 '17 at 21:38
  • Presumably you mean that is a semi circle, and the chords intersect on the circumference of the semicircle? – Matt Aug 18 '17 at 21:39
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    I understand this as: We have points $A,B,C,D$ on a circle (in this order), where $|AB|=3$, $|BC|=2$, $|CD|=1$ and $AD$ is a diameter. – Hagen von Eitzen Aug 18 '17 at 21:41
  • @HagenvonEitzen Can you verify that the image I have added is a correct understanding of the problem ? ... Anyone can you redraw this image ... nicely ? ... – Donald Splutterwit Aug 18 '17 at 23:00
  • You need to show what you have done to try and solve this problem. MSE usually looks unfavorably upon questions where no demonstrated attempt at a solution has been made. – Steven Alexis Gregory Aug 19 '17 at 12:22
  • Did you know that the perpendicular bisector of a chord of a circle passes through the center of the circle? – Steven Alexis Gregory Aug 19 '17 at 12:22

1 Answers1

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The length of a chord made in a circle radius $r$ by angle $\theta$ is $2r\sin\left(\dfrac{\theta}{2}\right)$

So radius $r$ is the solution to $$2\sin^{-1}\left(\frac{3}{2r}\right) + 2\sin^{-1}\left(\frac{2}{2r}\right)+2\sin^{-1}\left(\frac{1}{2r}\right) = \pi$$

There are apparently ways of adding arcsines, but in any case numerically it seems $r \approx 2.056545292$.

I have no idea whether it is connected, but this is the positive root of $2r^3-7r-3=0$.

This diagram shows the chords in the semicircle

enter image description here

Henry
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