Question

Represent $\left(1+\sqrt{9.5}\right)$ on the number line.

Answer 1

To represent $\left(1+\sqrt{9.5}\right)$ on the number line, follow the following steps of construction:

(i) Mark two points A and B on a given line such that AB = 9.5 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.

(vii) From E, mark a point F on the same given line such that EF = 1 unit.

Thus, let us treat the given line as the number line, with B as 0, C as 1, E as $\sqrt{9.5}$ and so on, then point F represents $\left(1+\sqrt{9.5}\right)$.


Justification:

Here, in semi-circle, radii OA = OC = OD = $\frac{9.5+1}{2}=\frac{{\displaystyle 10.5}}{2}=5.25\mathrm{units}$

And, OB = AB $-$ AO = 9.5 $-$ 5.25 = 4.25 units

In a right angled triangle OBD,

$$\begin{gathered} \mathrm{BD}=\sqrt{{\mathrm{OD}}^2-{\mathrm{OB}}^2} \\ =\sqrt{5.{25}^2-4.{25}^2} \\ =\sqrt{\left(5.25+4.25\right)\left(5.25-4.25\right)}\left[a^2-b^2=\left(a+b\right)\left(a-b\right)\right] \\ =\sqrt{9.5\times 1} \\ =\sqrt{9.5} \end{gathered}$$

$\mathrm{So},\mathrm{BF}=\mathrm{BE}+\mathrm{EF}=\sqrt{9.5}+1\left[\mathrm{BD}=\mathrm{BE}=\sqrt{9.5}\left(\mathrm{Radii}\right)\right]$