Perpendiculars and Parallels

≈ 36 min

Perpendiculars and Parallels

Three situations come up constantly, and each has its own construction.

A perpendicular at a point ON the line. From the point P, mark equal distances left and right along the line. You now have a short segment with P at its centre — bisect it. The bisector rises perpendicular through P.

A perpendicular from a point OFF the line. From the external point P, swing an arc that cuts the line twice. Those two cuts are equidistant from P, so bisecting the segment between them gives a line that passes through P at 90°.

A parallel at a set distance. Set the compasses to the required distance. Swing two arcs from any two points on the given line. A line drawn just touching the tops of both arcs — a tangent to both — is parallel at exactly that distance.

Core checkpoint: Every one of these reduces to bisecting something. Learn the bisection well and the rest follow.

When dropping a perpendicular from an external point P onto a line, what does the first arc swung from P achieve?

It marks two points on the line that are exactly the same distance from P.

It measures the shortest distance from P to the line directly.

It divides the line into equal parts.

It marks the midpoint of the line.

To construct a line parallel to a given line at a set distance, what is the correct method?

Set the compasses to that distance, swing arcs from two points on the line, and draw the line touching both arcs.

Measure the distance at one point only and draw a line through it by eye.

Bisect the given line and draw through the midpoint.

Draw any line and label it with the required distance.

Why is placing a set square by eye a poor substitute for constructing a perpendicular?

The placement error carries into every feature later located from that perpendicular.

Set squares cannot make 90° angles.

It takes longer than a compass construction.

Set squares are not permitted in EGD.

All three constructions in this lesson — perpendicular at a point, perpendicular from a point, and a parallel — reduce to repeated use of which single construction?