To find the square root of a positive real number geometrically
We show how to find
for any given positive real number 'x'geometrically.
For example, let us assume x = 4.5
We shall now find
geometrically.
PQ = 4.5 on a number line
Mark the distance 4.5 units from a fixed point P on a given line to obtain a point Q such that PQ = 4.5 units.
From Q, mark a distance of 1 unit and the new point as R.
Find the mid-point of PR and mark that point as O.
Draw a semicircle with centre O and radius OR.
Draw a line perpendicular to PR passing through Q and intersecting the semicircle at S.
Then, QS =
The square root of (4.5)
More generally,
to find
, for any positive real number x:
Mark Q so that
PQ = x units
Mark R so that QR = 1 unit. Refer to the figure below.
The square root of x
Then, as we have done for the case x = 4.5, we have QS =
We can prove this result using the Pythagoras Theorem.
From the figure, ΔOQS is a right-angled triangle. Also, the radius of the circle is
This shows that QS =
This construction gives us a visual, and geometric way of showing that
exists.
for all real numbers x > 0.
To represent
on a number line:
If you want to know the position of
on the number line,
then let us treat the line QR as the number line, with Q as zero, R as 1, and so on.
Draw an arc with center Q and radius QS, which intersects the number line in T.
The square root of x on a number line
Thus, T represents
.