Construct a perpendicular line through a point that does not lie on any line.

A line that makes a 90-degree angle with another line is called a perpendicular to that line.

Answer: Let's look into the construction of a perpendicular line through a point that does not lie on any line.

We need to make use of a compass and a ruler to construct the perpendicular line.

Explanation:

Given below are the steps of construction:

Given: A line and a point P that does not lie on the line.

• Adjust the compass in such a way that its opening is greater than the perpendicular distance from the point P to the line.
• Keeping the compass at point P, using this measure, scribe two arcs intersecting the line at two different points as shown below.

• Let the two points be A and B. Adjust the compass to a particular radius greater than half the length of the line segment AB and cut two arcs keeping the compass at points A and B respectively on the other side of the line opposite to point P.
• Let the point where the two arcs meet be Q as shown in the figure.

• Join the points P and Q as shown below.
• The angle made by PQ with AB is a right angle. Hence, PQ is perpendicular to AB through point P.

Hence, PQ is the line through point P perpendicular to AB.