Represent $\sqrt{7.128}$ geometrically on the number line.
To represent $\sqrt{7.28}$ on the number line, follow the following steps of construction:
(i) Mark two points A and B on a given line such that AB = 7.28 units.
(ii) From B, mark a point C on the same given line such that BC = 1 unit.
(iii) Find the mid point of AC and mark it as O.
(iv) With O as centre and radius OC, draw a semi-circle touching the given line at points A and C.
(v) At point B, draw a line perpendicular to AC intersecting the semi-circle at point D.
(vi) With B as centre and radius BD, draw an arc intersecting the given line at point E.
Thus, let us treat the given line as the number line, with $B$ as $0, C$ as 1 , and so on, then point $E$ represents $\sqrt{7.28}$.
Justification:
Here, in semi-circle, radii $\mathrm{OA}=\mathrm{OC}=\mathrm{OD}=\frac{7.28+1}{2}=\frac{8.28}{2}=4.14$ units
And, $\mathrm{OB}=\mathrm{AB}-\mathrm{AO}=7.28-4.14=3.14$ units
In a right angled triangle $\mathrm{OBD}$,
$\mathrm{BD}=\sqrt{\mathrm{OD}^{2}-\mathrm{OB}^{2}}$
$=\sqrt{4.14^{2}-3.14^{2}}$
$=\sqrt{(4.14+3.14)(4.14-3.14)} \quad\left[a^{2}-b^{2}=(a+b)(a-b)\right]$
$=\sqrt{7.28 \times 1}$
$=\sqrt{7.28}$