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Suppose that we have a line segment of length $8.4$ units. Is it possible to draw another line segment of length $8.4^2$ units, using only an unmarked straight edge and compass? I have no clue about how to approach this problem.

I could think of only one solution, first draw of square of side length $8.4$ units. Then squish the square, keeping its area constant, till one of its sides become unity, and hence the other side of the rectangle thus formed measures $8.4^2$ units. But the actions of the process cannot be performed under the given contraints.

Vansh Saini
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  • Do you also have a unit line segment? In other words, do you have $1$ and $x$, or just $x$? – Théophile Jul 02 '20 at 15:29
  • The ruler is unmarked. So there's no unit length segment to avail. – Vansh Saini Jul 02 '20 at 15:30
  • Do you request the segments to be aligned? – InsideOut Jul 02 '20 at 15:32
  • Yes, they can be aligned. – Vansh Saini Jul 02 '20 at 15:33
  • It is possible. 8.4 is a rational number, and segments with rational length are constructible. – user Jul 02 '20 at 15:35
  • Consider this: if $x < 1$, then you expect $x^2 < x$, whereas if $x > 1$, then $x^2 > x$. But without having a comparison to $1$, how would you know what to expect? – Théophile Jul 02 '20 at 15:35
  • To put it another way, it isn't meaningful to talk about a specific length like $8.4$ when that's the only length you have. You really need to have a base unit length somewhere first. – Théophile Jul 02 '20 at 15:42
  • @Théophile , Your arguement is reasonable. Suppose we have a unit length, then how to solve? – Vansh Saini Jul 02 '20 at 15:46
  • @Theophile You can multiply a given line segment to achieve any rational multiple of the original. Here the base is given as $8.4$ "units", all you are doing is multiplying it again by $8.4$, which is perfectly allowed. Your objection comparing the magnitudes of $x$ and $x^2$ falls through because you cannot square an arbitrary length $x$ without being told what its measure is. If it's given as $3$, I'd triple it, if it's given as $\frac 13$, I'd trisect it. Where's the issue? – Deepak Jul 02 '20 at 15:47
  • @VanshSaini In that case, the simplest way would be to draw a rectangle with sides $1$ and $x$. Scale up the rectangle by a factor of $x$; the new height is $x^2$. – Théophile Jul 02 '20 at 15:47
  • @Deepak I think the issue is the ambiguity of the question. Are we trying to multiply by a fixed number like $8.4$? If so, that's easy enough. But if we're trying to multiply by $x$, where $x$ is the length of the given segment, then there's a problem if we don't know how long it is. – Théophile Jul 02 '20 at 15:48
  • The length of the segment is already given, no need to consider an arbitrary length. – Vansh Saini Jul 02 '20 at 15:50
  • The problem is solved, thanks – Vansh Saini Jul 02 '20 at 15:51
  • @VanshSaini Thanks for clarifying. Here is the source of the confusion: there are two "kinds" of $8.4$ happening here. There's a length of $8.4$, and there's a ratio of $8.4$. Observe that it doesn't matter what the length was; your real question is just to scale any given line segment by the fixed ratio of $8.4$. – Théophile Jul 02 '20 at 15:58
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    @Theophile I have updated my answer and added a reference to your point, which I feel is a good one for the general problem. – Deepak Jul 02 '20 at 16:04

1 Answers1

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In principle all rational numbers are constructible.

You are given a length of $8.4$ and you're asked to multiply it by $8.4$. That's equivalent to multiplying it by $42$ then dividing it by $5$.

I'm sure you know how to multiply a line segment by extension and taking equal arcs on your compasses.

For the division, follow a method similar to this: https://youtu.be/CLMu6Wadux0

Just in case you're limited in the space (paper or screen) you can work with, you can also multiply the original segment by $8$, mark that off, then extend it by twice the original, then divide into fifths as before. That's basically $8 + \frac 25 = 8.4$ times the original.

Also, just for completeness, I will state that the point raised by Theophile is a valid one - you cannot square an arbitrary unknown length unless you are also provided another line segment of defined length (say $1$ unit). In your specific problem, that is not an issue as you are told that your original segment is $8.4$ units long.

Deepak
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