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Circle on a grid 1 (Posted on 2013-05-24) Difficulty: 3 of 5
Draw a unit circle on a sheet of graph paper with lines spaced 1 unit such that one square is completely inside the circle.

The lines of the paper will intersect the circle in 8 places, cutting the circle into 8 arcs. Consider the 4 arcs where some of the grid lines cut the circle as shown in the diagram: AB, BC, CD, DA.

Describe, as precisely as possible, how to place the circle so the relative lengths of these arcs are in the extended ratio 1:2:4:3 in order around the circle. Use the coordinate system as in the diagram.

No Solution Yet Submitted by Jer    
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Solution re(2): Not obvious to me Comment 3 of 3 |
(In reply to re: Not obvious to me by brianjn)

Let R be the centre of the circle of unit radius.  
The grid has graduations of 1 unit on the x and y
 
The arcs AB, BC, CD and DA subtend arcs of 36, 72, 144 and 108, these angles are in the required ratio of 1:2:4:3.
 ∠ARC=180
∠RCA=36
sin(36)=y/RC
y=RC*sin(36)
<br>
 ∠BRD=144
 ∠BDR=18
sin(18)=x/RD
x=RD*sin(18)

RC=RD=1 (radii)
x=sin(18)
y=sin(36)

R has the coordinates [x,y] or [sin(18),sin(36)].

I confirmed this result in a CAD drawing using a scale of 200 units (grid and radius) with a centre being [200*sin(18),200*sin(36)] and a Polar array of 10 radial segments around it.

  Posted by brianjn on 2013-05-28 23:13:25

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