Question

Show how \[\sqrt{5}\] can be represented on the number line.

Answer 1

Hint: First of all, take the length of the perpendicular BA as 1 unit and the length of the base OA as 2 units. Now, use the Pythagoras theorem for the \[\Delta BAO\] . According to the Pythagoras theorem we have, \[Hypotenuse=\sqrt{{{\left( Perpendicular \right)}^{2}}+{{\left( Base \right)}^{2}}}\] . Now, calculate the length of the hypotenuse. Then, by taking OB as the radius and O as the center, draw an arc intersecting the x-axis at point C. Now, the number \[\sqrt{5}\] is represented by the length of the line OC.

Complete answer:
According to the question, it is given that we have a number that is \[\sqrt{5}\] and we have to represent this number on the number line.
To represent the number \[\sqrt{5}\] , we need to use the Pythagoras theorem.
According to the Pythagoras theorem, we have
\[Hypotenuse=\sqrt{{{\left( Perpendicular \right)}^{2}}+{{\left( Base \right)}^{2}}}\] …………………………………………..(1)
We have to take the measure of the hypotenuse equal to \[\sqrt{5}\] units.
Now, taking 2 units as the base OA on the x-axis and a perpendicular BA of length 1 unit on that base.
Plotting OA = 2 units and the perpendicular BA = 1 unit on the number line, we get

In the \[\Delta BAO\] , we have
Perpendicular = BA = 1 unit …………………………………….(2)
Base = OA = 2 units ……………………………………….(3)
From equation (1), we have \[Hypotenuse=\sqrt{{{\left( Perpendicular \right)}^{2}}+{{\left( Base \right)}^{2}}}\] .
From equation (2) and equation (3), we have the lengths of the base and the perpendicular.
On putting, Perpendicular = BA = 1 unit and Base = OA = 2 units in equation (1), we get
\[\begin{align}
  & \Rightarrow Hypotenuse=\sqrt{{{\left( 1 \right)}^{2}}+{{\left( 2 \right)}^{2}}} \\
 & \Rightarrow Hypotenuse=\sqrt{1+4} \\
 & \Rightarrow Hypotenuse=\sqrt{5} \\
\end{align}\]
Now, take OB as a radius and draw an arc intersecting the real axis that is our x-axis at C.
Since \[\overset\frown{BC}\] is an arc that is drawn by taking O as the center so, we can say that the lengths OB and OC are equal because all the radii of an arc are equal to each other.
Therefore, \[OB=OC=\sqrt{5}\] .
So, OC represents the number \[\sqrt{5}\] on the real number line.
Hence, our number \[\sqrt{5}\] gets represented.

Note: In this question, one might forget to draw an arc by taking OB as the radius and conclude the length OB as the representation of the number \[\sqrt{5}\]. This is wrong because we have to represent \[\sqrt{5}\] on the real number line and we know that x-axis is the real number line. So, we have to draw an arc by taking OB as the radius which intersects the x-axis.