Lesson 3: Constructions and Loci – Using Ruler and Compass

🟦 Introduction

Hello there! 😊 In this lesson, we’ll explore the fascinating world of geometrical constructions and loci. You’ll learn how to draw accurate shapes and lines using only a ruler and a compass — just like mathematicians did before computers! Plus, we’ll see how loci (plural of locus) help us find all the possible places where something can be. Ready to draw, discover, and explore? Let’s begin!

🟩 Key Concepts and Explanations

🔹 What Are Geometrical Constructions?

Geometrical constructions are drawings made using only a ruler (to draw straight lines) and a compass (to draw arcs and circles). No measuring with protractors or scales — just pure accuracy through technique.

🛠 Common Constructions You Must Know

• Constructing a perpendicular bisector
• Bisecting an angle
• Drawing a perpendicular from a point to a line
• Constructing angles: 60°, 90°, 120° using compass

🔹 What Is a Locus?

A locus is a set of points that satisfy a certain rule or condition. For example, all points that are the same distance from a fixed point form a circle — that’s a locus!

🧭 Common Loci You Should Know

🟨 Sample Construction Walkthroughs

✏️ Example 1: Constructing a Perpendicular Bisector

Problem: Draw a line segment AB of length 6 cm and construct its perpendicular bisector.

Steps:

1. Draw AB = 6 cm using a ruler.
2. Place the compass on A and draw an arc above and below the line.
3. Keep the same radius, place compass on B, and draw arcs that cross the first ones.
4. Join the points where the arcs intersect. That line is the perpendicular bisector.

✏️ Example 2: Constructing the Locus of Points Equidistant from Two Points

Problem: Given two points P and Q, construct the locus of points equidistant from P and Q.

Steps:

1. Draw a line segment PQ.
2. Construct the perpendicular bisector of PQ (as in Example 1).
3. The perpendicular bisector is the locus you are looking for.

✏️ Example 3: Constructing the Locus of Points 2 cm from a Line

Steps:

1. Draw a horizontal line (label it AB).
2. Using a compass set to 2 cm, draw arcs above and below the line all along its length.
3. Join the arc edges with smooth lines — these are two lines parallel to AB, 2 cm away.

🟨 Practice Exercises

Section A: Construction Tasks

1. Construct a triangle XYZ where XY = 5 cm, YZ = 4 cm, and ∠XYZ = 60°. Use only ruler and compass.
2. Draw a line segment 8 cm and construct the perpendicular from a point 3 cm above the line to the line.

Section B: Loci Identification

1. What is the locus of points 4 cm from a fixed point?
2. What is the locus of points equidistant from two intersecting lines?
3. What is the locus of a point moving such that it stays 3 cm from line XY?

Section C: Matching

✅ Answers and Explanations

• Section A:
  1. Student should follow construction steps to complete triangle (diagram to be checked)
  2. Student should draw and drop a perpendicular using compass arcs (diagram to be checked)
• Section B:
  1. A circle of radius 4 cm
  2. The angle bisector of the two lines
  3. Two lines parallel to XY, 3 cm away
• Section C:
  1. Circle
  2. Two parallel lines
  3. Perpendicular bisector

🔁 Recap

We’ve explored how to make precise constructions with a ruler and compass, including bisectors and triangle building. You’ve also discovered how loci describe the path of points that follow specific conditions — like equal distance from a point or line. These are powerful tools in geometry, useful in both exams and real-life planning!

🪞 Reflection Prompt

Why do you think mathematicians prefer using ruler and compass for constructions? How might knowing loci help in real-life situations like engineering or landscaping?