#### (i) Represent geometrically the following numbers on the number line : $\sqrt{4\cdot 5}$ (ii) Represent geometrically the following numbers on the number line : $\sqrt{5\cdot 6}$ (iii) Presentation of $\sqrt{8\cdot 1}$ on number line : (iv) Presentation of $\sqrt{2\cdot 3}$ on number line:

(i) Solution. AB = 4.5 units, BC = 1 unit

OC = OD = $\frac{5\cdot 5}{2}$ = 2.75 units
OD2 = OB2 + BD2
$\left ( \frac{4\cdot 5}{2} \right )^{2}= \left ( \frac{4\cdot 5}{2} -1\right )^{2}+\left ( BD \right )^{2}$
$\Rightarrow BD^{2}= \left ( \frac{4\cdot 5+1}{2} \right )^{2}- \left ( \frac{4\cdot 5-1}{2} \right )^{2}$
$\Rightarrow BD^{2}= 4. 5$
$\Rightarrow BD= \sqrt{4. 5}$

So the length of BD will be the required one so mark an arc of length BD on number line, this will result in the required length.

(ii) Solution.  Presentation of $\sqrt{5. 6}$ on number line.
Mark the distance 5.6 units from a fixed point A on a given line to obtain a point B such that AB = 5.6 units. From B mark a distance of 1 unit and mark a new point C. Find the mid point of AC and mark that point as O. Draw a semicircle with center O and radius OC. Draw a line
perpendicular to AC passing through B and intersecting the semicircle at O. Then BD = $\sqrt{5\cdot 6}$

(iii) Solution
Mark the distance 8.1 units from a fixed point A on a given line to obtain a point B such that AB = 8.1 units. From B mark a distance of 1 unit and mark the new point AB. Find the mid point of AC and mark a point as O. Draw a semi circle with point O and radius OC. Draw a line perpendicular to AC passing through B and intersecting
the semicircle at D. Then BD -
$\sqrt{8. 1}$

(iv) Solution
Mark the distance 2.3 unit from a fixed point A on a given line. To obtain a point B such that AB = 2.3 units. From B mark a distance of 1 unit and mark a new point as C. Find the mid point of AC and mark the point asO. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then BD = $\sqrt{2. 3}$