3

In plane geometry the product of two line segments p and q can be represented as the area of a rectangle with sides p and q. Or at least that is the premise assumed here. Assuming that is correct, the question is: Can this product be represented by a simple line segment, instead of an area?

bubba
  • 43,483
  • 3
  • 61
  • 122
William Falcon
  • 31

1 Answers1

6

Yes it is possible.

We are given two line segments, of lengths $a$ and $b$. Draw two say perpendicular lines (it doesn't really matter) meeting at some point $O$. On one of the lines, which we call the $x$-axis, make a point $A$ such that $OA=a$. (Straightedge and compass can do this.) On the other line, which we call the $y$-axis, make a point $B$ such that $OB=b$.

On the $x$-axis, put a point $X$ such that $OX$ has unit length. Join $X$ and $B$.

Through $A$, draw the line parallel to $XB$. This meets the $y$-axis at some point $P$.

By similar triangles, we have $\frac{b}{1}=\frac{OP}{a}$. It follows that $OP$ has length $ab$.

Remark: By a small modification of the basic idea, we can also construct a line segment of length $\dfrac{a}{b}$.

Note that we need to define, perhaps arbitrarily, some line segment as the unit line segment in order to carry out the construction.

André Nicolas
  • 507,029
  • I would highlight that this application of the Thales' theorem or any other method one could devise requires defining the unit, as opposed to the area representation, because scaling a and b by k would scale the segment by k*k and not by k, so the relative length of the product segment (the proportions of the constructed triangle) is scale dependent, if I'm not wrong. – Andrestand Jan 29 '21 at 11:26
  • The division, on the other hand, scales properly, but you require to define a unit length to which refer the decimal part. – Andrestand Feb 10 '21 at 13:03