Multiplication of lines of irrational length

Question

Suppose you are given two lines, each has length of an irrational number. How would one, using a straight-edge and compass, draw a line whose length is the product of the two given irrational numbers?

Edit: Yes, you are also given a unit line. But note that the irrational numbers are not necessarily square roots.

Are we given a unit line? I guess not. I feel a little ... sucker punched ... being told the lines are irrational without a unit reference. — fleablood, Jul 03 '16 at 16:46
If the numbers are equal I'd think it would be pretty easy to construct the square. Can you be a bit more specific? In what context is this? — John Molokach, Jul 03 '16 at 16:59
It cannot be done without a unit length, or something equivalent. With a unit length one uses similar triangles. — André Nicolas, Jul 03 '16 at 17:58

score 2 · Accepted Answer · answered Jul 27 '16 at 14:09

2

Hint: Use similar triangles and the identity $\dfrac{rs}{r} = \dfrac{s}{1}$. You have 3 of the terms and want the 4th.

answered Jul 27 '16 at 14:09

user21820

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Note to the OP: the fact that the original lengths are irrational is irrelevant. – Ethan Bolker Jul 27 '16 at 14:12

score 1 · Answer 2 · answered Jul 27 '16 at 14:18

1

Construct a segment $\overline{ACB}$ such that $AC = r$ and $CB = s$.

Construct a perpendicular through $C$, and mark off a point such that $CD = 1$.

Construct the circle $O$ through $A, B,$ and $D$.

Find the intersection point $E$ of $O$ and $\overline{CD}$.

Then by the power of a point theorem, $CE = rs$.

answered Jul 27 '16 at 14:18

Yup this is a nice way too. – user21820 Jul 27 '16 at 14:34

Multiplication of lines of irrational length

2 Answers2