Question

The perpendicular drawn on a perpendicular of a line is parallel to the initial line. Is this statement always true ? Justify your answer.

Answer 1

Yes, the given statement is always true.

Step $1$: Drawing first perpendicular :

Let us draw a perpendicular on a line using a ruler and compasses :

Step $A$: Draw a line $n$ and take a point $O$ on it. Taking a radius of any length, draw a semicircle with $O$ as the centre. The semicircle cuts the line $n$ at two points, $P$ and $Q$.

Step $B$: With $P$ as the centre and a radius of any length greater than $PO$, draw an arc on one side of the line $n$ over the semicircle.

Step $C$: Now taking the same radius as before and $Q$ as the centre, draw another arc intersecting the previous one at $R$.

Step $D$: Join the points $O$ and $R$, and extend the line. Here, the line segment $OR$ is perpendicular to line $n$.

Step $2$: Drawing second perpendicular (A perpendicular on perpendicular) :

Let us draw another perpendicular on the previous perpendicular $OR$, using a ruler and compasses :

Step $A$: Taking a radius of any length, draw a semicircle with $A$ as the centre. The semicircle cuts the line segment $OR$ at two points, $B$ and $C$.

Step $B$: With $B$ as the centre and a radius of any length greater than $BC$, draw an arc on one side of the line segment $OR$ over the semicircle.

Step $C$: Now taking the same radius as before and $C$ as the centre, draw another arc intersecting the previous one at $D$.

Step $D$: Join the points $A$ and $D$, and extend the line. Here, the line segment $AD$ is another perpendicular to line segment (first perpendicular) $OR$. Thus, the second perpendicular $AD$ is parallel to the initial line.

Hence, the given statement the perpendicular drawn on a perpendicular of a line is parallel to the initial line, is always true.