How I construct a segment $a^2$?

Question

The length of a segment is given. How can we construct a segment equal to the square of the given segment?

Okey, but how do I construct a^2 if I have been given the length of "a". — Maxim Kasnedelchev, May 13 '18 at 16:43
Do you mean constructing with the given straight segment and a compass? — DonAntonio, May 13 '18 at 16:44
Yes I should construct it with a compass and a straight segment. — Maxim Kasnedelchev, May 13 '18 at 16:48

score 4 · Accepted Answer · answered May 13 '18 at 16:42

4

Use this construction. BTW, it can also be used to multiply $a$ times $b$:

The green lines are parallel. The blue lines are any two that form an angle. You need a unit length as well, of course.

answered May 13 '18 at 16:42

John Hughes

What app did you use for the diagram? – Maxim Kasnedelchev May 13 '18 at 16:45
1

But then this construction requires two line segments: one of length $;a;$ , the other one of length $;1;$ ... – DonAntonio May 13 '18 at 17:22
That's what "You need a unit length as well, of course" meant. :) – John Hughes May 13 '18 at 18:56
The "app" I used was ... PowerPoint. :) – John Hughes May 13 '18 at 18:57
Also: see silverfish's comment on the question here (https://math.stackexchange.com/questions/705/compass-and-straightedge-construction-of-the-square-root-of-a-given-line) to see why a unit length is necessary for the question even to make sense. (Also Isaac's answer to the same question.) – John Hughes May 13 '18 at 19:23

score 0 · Answer 2 · answered May 13 '18 at 16:45

0

Carry out the inversion of the point $1$ on the $x$-axis with respect to the circle of radius $a$ centered at the origin.

answered May 13 '18 at 16:45

Mikhail Katz

Peter Szilas · Answer 3 · 2018-05-13T22:00:18.597

0

Revised attempt:

1) Construct a $\triangle DBC$ with :

$|DB|=1;$ $ \angle BDC =90°$; $|DC|=a$ .

2) Construct a right angle at $C$, with one leg $BC$.

The other leg of this angle intersects $BD$

at $A$, I.e $\angle BCA =90°$.

3) We have :

$|AD| \cdot 1 = a^2.$

edited May 13 '18 at 22:00

answered May 13 '18 at 17:29

Peter Szilas

John.Oops, you are right.My mistake! – Peter Szilas May 13 '18 at 19:41
John Hughes.Hopefully ok now. Thanks for your comment. – Peter Szilas May 13 '18 at 22:07
Looks good! Note that your solution (like mine) requires that you have a unit length as well as the length $a$ (see DonAntonio's comment on my answer). – John Hughes May 13 '18 at 22:13
John Hughes.Thanks, also for pointing out Don Antonio's comment. – Peter Szilas May 13 '18 at 22:32
Could it be done without a given unit length? – Maxim Kasnedelchev May 14 '18 at 18:01
Maxim.John Hughes' and Don Antonio's comment, you need a length a, and a unit length.My answer: Under 1) :construction of triangle DBC we use Side, Angle,Side with lengths a,1, and angle 90°.ok? – Peter Szilas May 14 '18 at 18:20

3 Answers3