A perpendicular line meets another line at a right angle of 90°. Constructing one with only a compass and straightedge is a classic geometry skill because it uses equal distances rather than measuring with a protractor. This makes the result exact in ideal geometric reasoning.
The construction also shows how circles, arcs, and symmetry can create precise angles.
Understanding Geometry: Constructing a Perpendicular Line
Compass constructions are really arguments about distance. A drawn arc is not just a curved mark. It records every location that is one chosen distance from its center.
When two arcs cross, each crossing has a useful distance relationship to two earlier points. This is why the construction works without reading any scale. The final straight line is supported by symmetry.
It goes through places that cannot favor one endpoint of a segment over the other. In geometry, this idea is called a locus.
A locus is the full set of points that satisfy one rule. The points equally far from two fixed endpoints form a straight path through the segment's middle.
The two situations need slightly different planning. When the given point lies on the original line, the construction must first create a segment centered at that point. The endpoints should be placed on opposite sides, with matching compass widths.
This makes the given point the midpoint of the temporary segment. When the given point is away from the line, an arc from that point creates the two endpoints where it crosses the line. Those endpoints lie at the same distance from the outside point automatically.
In both cases, the temporary segment is the important object. The desired line comes from finding its perpendicular bisector.
A proof of the result uses congruent triangles. Take either intersection of the later arcs and connect it to the two temporary endpoints. The two sides made by those arcs have equal length because the compass opening did not change.
The distances from the given point to the two endpoints are equal as well. The two triangles share the segment between the given point and the arc intersection. Their matching side lengths show that the triangles are congruent.
This forces the angles at the midpoint to match. Since those matching angles sit next to each other on a straight line, together they make a straight angle.
Equal halves of a straight angle are right angles. This is the reason the construction is exact in ideal geometry.
Small drawing errors can hide the structure, even when the reasoning is correct. Keep the compass point steady and make the arcs long enough to cross clearly. Do not change the compass width while making a pair of arcs meant to be equal.
Use a sharp pencil because thick lines make intersections hard to locate. A straightedge should guide the final line only, not measure distances. Students often try to force the final line through a visually guessed right angle.
Instead, trust the intersection points created by equal distances. Perpendicular layouts appear in graph paper, building plans, road grids, coordinate axes, and technical drawings. The same distance logic later supports constructions of angle bisectors, triangles, and regular polygons.
Key Facts
- Perpendicular lines meet at a right angle: m∠ABC = 90°.
- A compass keeps a fixed radius, so points on the same arc are the same distance from the center.
- The perpendicular bisector of segment CD is the set of points equidistant from C and D.
- For a point P on line l, choose points A and B on l so that PA = PB, then construct the perpendicular bisector of AB.
- For a point P off line l, draw an arc centered at P that intersects l at A and B, then construct the perpendicular bisector of AB.
- If PA = PB and QA = QB, then line PQ is perpendicular to AB.
Vocabulary
- Perpendicular lines
- Two lines that intersect to form a 90° angle.
- Compass
- A drawing tool used to make circles or arcs with a fixed radius.
- Straightedge
- A tool used to draw a straight line through two points without measuring length.
- Arc
- A connected part of a circle drawn by a compass.
- Perpendicular bisector
- A line that cuts a segment into two equal parts and meets it at a 90° angle.
Common Mistakes to Avoid
- Changing the compass width between matching arcs, which breaks the equal-distance relationships needed for the construction to be exact.
- Placing the off-line point too close to the given line for the chosen compass radius, which may prevent the arc from crossing the line in two usable points.
- Drawing the final line through the wrong arc intersection, which can create a slanted line that is not perpendicular to the original line.
- Using a ruler scale or protractor instead of compass and straightedge steps, which turns an exact construction into an approximate measurement.
Practice Questions
- 1 A point P lies on line l. You mark points A and B on l so that PA = 4 cm and PB = 4 cm. If arcs centered at A and B meet at Q, what line should you draw to construct the perpendicular to l through P?
- 2 Point P is 3 cm above line l. You draw an arc centered at P with radius 5 cm, and it meets line l at A and B. If PA = PB = 5 cm, what construction must be done next to find the perpendicular from P to l?
- 3 Explain why the perpendicular constructed from a point off a line passes through the original point P when the same compass radius is used to mark equal distances to two points on the line.