How to Construct a 1 Degree Arc Using a Given 19 Degree Arc?

[mishaark]

Exercise #27 from a textbook called Kiselev's Geometry / Book I. Planimetry:

Using only compass, construct a 1 degree arc on a circle, if a 19 degree arc of this circle is given.Please, check my reasoning on this one. I just want to make sure that I'm getting it right.

My solution:
Using a pair of compasses, we can only divide an angle in half. So long as we are given a 19 degree arc, we cannot really apply this method because we will end up having angles with fractional parts. But if we take a 19 degree angle 10 times we will get the angle of 190 degrees which we can divide in half and get an 80 degree angle, which in turn divided in half will give us a 40 degree angle, which again divided in half will give us a 20 degree angle. Now we can superimpose this 20 degree angle onto the 19 degree angle, which will give us by subtraction a 1 degree angle, that is a 1 degree arc.

 

[tiny-tim]

welcome to pf!

hi mishaark! welcome to pf!

(have a degree: ° )

… we will get the angle of 190 degrees which we can divide in half and get an 80 degree angle …

you haven't actually said how you get from 190° to 80°

(but isn't there an easy way of getting 18° ?)

 

[coolul007]

You can make a 15 degree angle also by dividing a 60 degree angle twice then subtract it from 19. Then divide the remainder twice.

 

[Alexander Glauberzon]

You can make a 15 degree angle also by dividing a 60 degree angle twice then subtract it from 19. Then divide the remainder twice.

In order to divide angle by 2 required compass and straight, thus none of the solutions above will satisfy condition to use compass only.

My solution is to draw circle with center at the vertex of the angle. Using compass measure chord of 19 degrees and add it to the original chord 18 times the result angle is 19*19 = 361 degrees from original point. The rest is obvious.

 

[coolul007]

According to this Mohr–Mascheroni theorem one can use compass only if compass and straight edge can be constructed.

 

[olivermsun]

In order to divide angle by 2 required compass and straight, thus none of the solutions above will satisfy condition to use compass only.

My solution is to draw circle with center at the vertex of the angle. Using compass measure chord of 19 degrees and add it to the original chord 18 times the result angle is 19*19 = 361 degrees from original point. The rest is obvious.

Yup, I think you have the correct solution. For construction the problem gives you a 19° arc of the circle, so all you have to do is "copy" the arc around the circle with your compass.