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.
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.