Constructions | Class 7 Mathematics Notes

Constructing a Perpendicular to a Line from a Point on the Line

This section describes how to construct a perpendicular line to a given line l from a point P lying on the line. The construction is important in many geometric problems and is done using a compass and straightedge. The steps are as follows: first, place the compass pointer at point P and draw arcs intersecting the line l at two points, say A and B, on either side of P. Next, with the compass pointer at A and radius more than half the distance AB, draw an arc above the line. Repeat the same from point B with the same radius, creating an intersection point C above the line. Finally, draw a straight line from P through point C. This line is the required perpendicular to line l at point P. The section explains the reasoning behind this construction, noting that triangle PAC and PBC are congruent, ensuring the angle at P is 90°. This method is precise and does not require measuring angles with a protractor. The section also discusses the importance of perpendicular lines in geometry, such as in constructing squares, rectangles, and right angles.

📊 Diagram: The diagrams illustrate line l with point P on it. Arcs from P intersect l at A and B. Arcs from A and B intersect at C above the line. The perpendicular line PC is drawn, forming a right angle with l at P.

🧪 Activity: Activity: Using a compass and ruler, construct a perpendicular to a given line from a point on the line and verify the right angle using a set square.

🔗 Connection: This section leads to constructing perpendiculars from a point outside the line, which is covered next.