Welcome to Loci and Constructions!

Have you ever wondered how architects ensure a building is perfectly symmetrical, or how a gardener decides where to place a sprinkler so it hits every corner of a lawn? That is exactly what we are exploring today! In this chapter, we move away from just "sketching" and focus on precise drawing using mathematical tools.

Note: In your Edexcel IGCSE exam, you must use a ruler and compasses for these tasks. The syllabus specifically states that tracing paper methods are not acceptable for loci. So, grab your geometry set and let’s get started!

1. What is a Locus?

The word locus (the plural is loci) sounds complicated, but it simply means a set of points that follow a specific rule. Think of it as the path left behind by a moving object.

Analogy: Imagine a dog tied to a post by a 3-metre leash. If the dog walks as far as it can around the post, the path it makes is a circle. That circle is the "locus" of the dog!

Common Loci You Need to Know:

  • From a fixed point: The locus of points at a distance \(d\) from a point \(P\) is a circle with radius \(d\) and centre \(P\).
  • From a straight line: The locus of points at a fixed distance from a line is a pair of parallel lines (often with rounded "caps" at the ends if the line is a segment).
  • Equidistant from two points: This is a straight line exactly halfway between them (the perpendicular bisector).
  • Equidistant from two lines: This is a line that cuts the angle between them exactly in half (the angle bisector).

Quick Tip: If a question says "equidistant," it just means "at the same distance."

2. Construction: The Perpendicular Bisector

This is used when you need to find all the points that are exactly the same distance from two fixed points, let's call them \(A\) and \(B\). This line is also called the mediator.

Step-by-Step Guide:

1. Place your compass point on point \(A\).
2. Open your compass so it is more than half the distance to \(B\).
3. Draw an arc above and below the line \(AB\).
4. Without changing the compass width, move the point to \(B\) and draw two more arcs that cross your first ones.
5. Use your ruler to draw a straight line through the two points where the arcs cross.

Key Takeaway: Every point on this new line is exactly the same distance from \(A\) as it is from \(B\). The line also hits \(AB\) at a \(90^{\circ}\) angle!

3. Construction: The Angle Bisector

This construction is used to split an angle perfectly in half. It creates a line where every point is the same distance from the two lines forming the angle.

Step-by-Step Guide:

1. Place the compass point on the vertex (the corner) of the angle.
2. Draw an arc that crosses both lines of the angle.
3. Place the compass point on one of the spots where the arc crosses a line, and draw a small arc in the middle of the angle.
4. Without changing the compass width, repeat this from the other crossing point so the two small arcs intersect.
5. Use a ruler to join the vertex of the angle to the point where the arcs crossed.

Did you know? This is how you find the path that stays exactly halfway between two diverging roads!

4. Shading Regions (Regions of Loci)

Sometimes, the exam will ask you to shade a specific area (a region) that satisfies several rules at once. This is where you combine your constructions.

Example Problem:

"Shade the region that is less than \(5\) cm from point \(A\) AND closer to point \(B\) than point \(C\)."

How to solve this:

1. Rule 1: Draw a circle (or arc) with a radius of \(5\) cm around point \(A\). The region "less than" will be inside this circle.
2. Rule 2: Construct the perpendicular bisector between \(B\) and \(C\). This line is the "border."
3. Decide: Which side of the border is closer to \(B\)? (It’s the side \(B\) is on!)
4. Shade: Find the area that is both inside the circle AND on the \(B\) side of your bisector line.

Common Mistake to Avoid: Don't rub out your construction arcs! The examiner needs to see them to give you marks for your method. Keep them light but visible.

5. Summary and Quick Review

To succeed in this chapter, remember these three "golden rules" of construction:

  • Don't Fiddle: Once you set your compass width for a specific construction step, do not let it slip or change it until that step is done.
  • Keep it Sharp: Use a sharp pencil. A blunt lead makes thick lines that lead to "accuracy errors" in exams.
  • Read the wording: "From a point" usually means a circle. "From a line" usually means parallel lines. "From two points" means a perpendicular bisector.

Final Check: Do you have your ruler, compasses, and protractor ready? (While we use a ruler and compasses for constructions, a protractor is often listed in the "mathematical instruments" you might need for this section to check your work or solve related angle problems).

Don't worry if your arcs look a bit messy at first—precision comes with practice. Just remember: keep those construction lines visible, and you'll pick up those marks!