Question

Represent $\sqrt{5},\sqrt{6}$ and $\sqrt{7}$ on the number line.

Answer 1

Step 1: Represent $\sqrt{5}$ on the number line:

1. Draw a horizontal line $X^{\text{'}}OX$, taken as the $x$-axis. Take $O$ as the origin to represent $0$.
2. Let $OA=2$ units and $AB\perp OA$ such that $AB=1$ unit.

3. Join $OB$ and get $OB=\sqrt{O{A}^2+A{B}^2}=\sqrt{2^2+1^2}=\sqrt{5}$.

4. With $O$ as center and $OB$ as radius, draw an arc, meeting $OX$ at $P$.
So, $OP=OB=\sqrt{5}$. Thus, $P$ represents $\sqrt{5}$ on the number line.

Step 2: Represent $\sqrt{6}$ on the number line

1. Draw $BC\perp OB$ and take $BC=1$ unit.
2. Join $OC$ and get $OC=\sqrt{O{B}^2+B{C}^2}=\sqrt{(\sqrt{5}{)}^2+1^2}=\sqrt{6}$ .
3. With $O$ as center and $OC$ as radius, draw an arc, meeting $OX$ at $Q$.

Thus, $Q$ represents $\sqrt{6}$ on the number line.

Step 3:Representing $\sqrt{7}$ on the number line

1. Draw $CD\perp OC$ and take $CD=1$ unit.
2. Join $OD$. and get $OD=\sqrt{O{C}^2+C{D}^2}=\sqrt{(\sqrt{6}{)}^2+1^2}=\sqrt{7}$ .
3. With $O$ as center and $OD$ as radius, draw an arc, meeting $OX$ at $R$.

Here, $OR=OD=\sqrt{7}$. Thus, R represents $\sqrt{7}$ on the number line.

Hence, on the number line, $P,QandR$ represent $\sqrt{5},\sqrt{6}$ and $\sqrt{7}$ respectively .