Show, how given a distance r, one can construct a regular pentagon, whose circumradius is r, using only ruler and compass...
Note 1: The ruler doesn't have any marks, so it's no good for measuring. It's only good for joining 2 points by a line.
Note 2: The circle which circumscribes the polygon, such that the polygon lies entirely within the circle and all of whose vertices lie on the circumference of the circle is the circumcircle of the polygon. The radius of this circle is the circumradius of the circle.
Given the line segment of length 'r'. From one end of the line segment make an angle of 72 degrees with the compass. Then draw the straight line from that point such that the angle between the two lines (the given and the one that will be drawn) will be equal to 72 degrees. Then using the compass take the measure of the distance 'r' from the given line and cut an arc which cuts the second line at a distance 'r'. This is the second radius of the pentagon. Continuing the same procedure four more times, we will get six such straight lines such that when taken in pair, the angle between every two consecutive straight lines will be equal to 72 degrees and each straight line will be of length 'r'. Now join all the end points of these straight lines and from the common point (from where the angles were measured and arcs of length 'r' were cut off), draw a circle of radius 'r'. We now get a circumcircle of radius 'r' circumscribing a regular polygon.
(I think I have not framed my answer properly but I hope everyone understands what I am trying to say. I mean my method. Actually the moment I thought of it I just posted everything out here).
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