CLASS-6CONSTRUCTION OF PERPENDICULAR TO A LINE FROM A POINT OUTSIDE THE LINE

CONSTRUCTION OF PERPENDICULAR TO A LINE FROM A POINT OUTSIDE THE LINE -

Constructing a perpendicular line to a given line from a point outside the line involves several steps. Here's how you can do it using a compass and a straightedge:

Let's say you have a line ℓℓ and a point P outside that line, and you want to construct a line perpendicular to ℓℓ passing through point P.

1. Draw the given line ℓℓ and mark the point P outside it.
2. Place the compass on point P:- Put the compass needle on point P.
3. Adjust the compass width:- Adjust the compass width to a length longer than the distance from point P to the line ℓℓ.
4. Draw an arc:- With the compass set at the adjusted width, draw an arc that intersects the line ℓℓ at two points. Label these points of intersection as A and B.
5. Place the compass on A and B:- Without changing the compass width, place the compass needle on points A and B successively.
6. Draw arcs:- With the compass set at the same width, draw arcs with centers at points A and B such that they intersect above and below point P. Label these points of intersection as C and D.
7. Connect P with C and D:- Use a straightedge to draw lines from point P to points C and D.
8. Draw the perpendicular line:- The line PC or PD is the perpendicular line to ℓℓ passing through point P.

The rationale behind this construction is that the perpendicular distance from the given line ℓℓ to point P is the shortest distance. By using arcs to find equidistant points on the line, we ensure that the lines PC and PD are perpendicular to ℓℓ.

Another Way Of Understanding:-

To draw a perpendicular to a line from a point outside the line.

Given:- A line AB and a point P outside AB.

Required:- To draw a perpendicular to AB from the point P.

Steps Of Construction:-

Step.1) With P as centre and any suitable radius, draw an arc to cut the line AB at points C & D.

Step.2) With C & D as centres, draw two arcs of equal radius (>1/2 CD) cutting each other at Q on the other side of AB.

Step.3) Draw a line through P & Q to intersect the line AB at N, then segment PN is the required perpendicular from the point P to the line AB.