Lesson 13.1.2

Constructing Perpendicular & Angle Bisectors

The perpendicular bisector of a segment and the bisector of an angle are two of the most important constructions in geometry.

Introduction

Past Knowledge

Copying segments/angles (13.1.1). Properties of bisectors (1.3). Congruent triangles.

Today's Goal

Construct a perpendicular bisector and an angle bisector.

Future Success

Parallel lines (13.1.3), inscribed polygons (13.1.4–5), circumcenters & incenters.

Key Concepts

Perpendicular Bisector

A line that is both ⊥ to a segment AND passes through its midpoint. Every point on it is equidistant from the segment's endpoints.

Angle Bisector

A ray that divides an angle into two congruent angles. Every point on it is equidistant from the two sides.

Step-by-Step Constructions

Perpendicular Bisector of

  1. 1Open compass to more than half of AB.
  2. 2From A, draw arcs above and below the segment.
  3. 3Without changing compass width, from B draw arcs crossing the first ones.
  4. 4Connect the two intersection points → perpendicular bisector.

Angle Bisector of ∠ABC

  1. 1From vertex B, draw an arc crossing both sides of the angle.
  2. 2From each intersection point, draw equal arcs that cross each other.
  3. 3Draw a ray from B through the intersection → angle bisector.

Why It Works

Perpendicular Bisector: The two arcs from A and B with equal radii create a rhombus (all 4 segments = compass width). A rhombus's diagonals are perpendicular bisectors of each other → QED.

Angle Bisector: The construction creates two congruent triangles (SSS: equal arc radii), so the corresponding angles are equal → bisected.

∎ Both rely on SSS congruence from equal compass settings.

Common Pitfalls

Compass Too Narrow

For the perpendicular bisector, the compass must be open to MORE than half the segment — otherwise the arcs won't intersect.

Real-Life Applications

Finding the Center of a Circle

Construct perpendicular bisectors of any two chords — they intersect at the center. This works even for broken circular objects.

Map Navigation

The perpendicular bisector between two locations gives all points equidistant from both — useful for finding optimal meeting points.

Practice Quiz

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