Question
Represent $\sqrt{6},\sqrt{7},\sqrt{8}$ on the number line.
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Answer
Draw a number line and mark a point $O$, representing zero, on it.
Suppose point A represents $1$ as shown. Then, $O A=1$.
Draw a right triangle $O A B$ such that $A B=O A=1$.
By pythagoras theorem, we have
$(OB)^2 = (OA)^2 +(AB)^2$
$\Rightarrow OB^2 = 1^2 + 1^2 = OB2 = 1 +1 = 2$
$\Rightarrow\text{OB}=\sqrt{2}$
Now, draw a circle with centre $O$ and radius $O B$. We find that the arcle cuts the number line at $A$.
Clearly, $A _1= OB =$ Radius of the cirde $=\sqrt{2}$
Thus, $A_1$ represents $\sqrt{2}$ on the number line.
Now, draw a right triangle $OBB _1$, such that $BB _1=2$.
Again by pythagoras theorem, we have,
$\text{OB}^2_1=\text{OB}^2+\text{BB}^2_1$
$\Rightarrow\text{OB}^2_1=\big(\sqrt{2}\big)^2+(2)^2$
$\Rightarrow\text{OB}^2_1=6$
$\Rightarrow\text{OB}_1=\sqrt{6}$
Now, draw a circle with centre $O$ and radius $OB _1$. We find that the circle cuts the number line at $A _2$.
Clearly, $OA _2= OB _2=$ Radius of circle $=\sqrt{6}$
Thus, $A_2$ represents $\sqrt{6}$ on the number line.
Now, draw a right angle triangle $OB _1 B_2$ such that $B _1 B_2=1$.
By pythagoras theorem, we have,
$\text{OB}^2_2=\text{OB}^2_1+\text{B}_1\text{B}^2_2$
$\Rightarrow\text{OB}^2_2=\big(\sqrt{6}\big)^2+(1)^2$
$\Rightarrow\text{OB}^2_2=6+1=7$
$\Rightarrow\text{OB}_2=\sqrt{7}$
Now, draw a circle with centre $O$ and radius $OB _2$. We find that the circle cuts the number line at $A _3$.
Clearly, $OA _3= OB _2=$ Radius of circle $=\sqrt{7}$
Thus, $A_3$ represents $\sqrt{7}$ on the number line.
Now, again draw a right triangle $OB _2 B_3$ such that $B _2 B_3=1$.
By pythagoras theorem, we have,
$\text{OB}^2_3=\text{OB}^2_2+\text{B}_2\text{B}^2_3$
$\Rightarrow\text{OB}^2_3=\big(\sqrt{7}\big)^2+(1)^2$
$\Rightarrow\text{OB}^2_3=7+1=8$
$\Rightarrow\text{OB}_3=\sqrt{8}$
Now, draw a circle with centre $O$ and radius $OB _3$. We find that the circle cuts the number line at $A _4$.
Clearly, $OA _4= OB _3=$ Radius of circle $=\sqrt{8}$
Thus, $A_4$ represents $\sqrt{8}$ on the number line.
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