Introduction to Constructions, Loci and Bearings

Welcome to one of the most practical chapters in GCSE Mathematics! In this section of Geometry and measures, we move away from just calculating numbers and start drawing with precision. Whether you are navigating a ship using bearings or finding the perfect spot for a new Wi-Fi router using loci, these skills are used every day by architects, engineers, and pilots.

Don't worry if you don't feel "artistic"—these topics are about following logical steps with the right tools. If you can use a pair of compasses and a ruler, you can master this!

Section 1: The Essential Toolkit

To succeed in this chapter, you must have your "maths set" ready. For the exam, you will need:

  • A sharp pencil: Thick lines lead to inaccurate drawings and lost marks!
  • A ruler: For straight lines and measuring distances.
  • Compasses: Used for drawing arcs and circles. Make sure the hinge is tight.
  • A protractor: Essential for measuring bearings.

Common Mistake: Never draw constructions freehand. The examiner looks for construction arcs (the little curved lines made by your compasses). Do not rub them out!

Section 2: Ruler-and-Compass Constructions

There are three "standard" constructions you must know. These are the building blocks for more complex problems.

1. Perpendicular Bisector

This is used to find the exact middle of a line at a \(90^{\circ}\) angle. "Perpendicular" means at a right angle, and "Bisector" means to cut into two equal parts.

  1. Place the compass needle on one end of the line (Point \(A\)).
  2. Open the compasses so the width is more than half the length of the line.
  3. Draw an arc above and below the line.
  4. Keeping the compass width the same, move the needle to Point \(B\) and draw arcs that cross your first ones.
  5. Use a ruler to join the two points where the arcs cross.

2. Angle Bisector

This cuts any angle exactly in half.

  1. Place the compass needle on the vertex (the corner) of the angle.
  2. Draw an arc that crosses both lines forming the angle.
  3. Place the needle on one of the points where the arc crosses a line and draw an arc in the middle of the angle.
  4. Repeat this from the other crossing point (keep the compass width the same!).
  5. Use a ruler to join the vertex to the point where these two new arcs cross.

3. Perpendicular from a Point to a Line

This finds the shortest distance from a point to a line. The shortest distance is always a straight line at \(90^{\circ}\).

  1. Place the needle on the point (let's call it Point \(P\)).
  2. Open the compasses wide enough to draw an arc that crosses the line in two places.
  3. From those two new crossing points, use the "Perpendicular Bisector" method to find the middle.
  4. Connect Point \(P\) to that middle point.

Quick Review: "Bisect" = Cut in half. "Perpendicular" = \(90^{\circ}\) angle.

Section 3: Loci (The Plural of Locus)

A locus is a set of points that follow a specific rule. Think of it as a "boundary" or a "path."

Common Types of Loci:

  • Rule: "A fixed distance from a point"
    Result: A circle. (Example: A dog on a \(3\text{m}\) leash can move anywhere within a circle of radius \(3\text{m}\)).
  • Rule: "A fixed distance from a straight line"
    Result: Two parallel lines (one on each side) with rounded "caps" at the ends.
  • Rule: "Equidistant from two points"
    Result: The perpendicular bisector of the line connecting the two points.
  • Rule: "Equidistant from two lines"
    Result: The angle bisector between the two lines.

Region Problems: Often, you are asked to shade a region. For example, "Shade the region closer to Point \(A\) than Point \(B\) and less than \(5\text{cm}\) from Point \(C\)." You would draw the perpendicular bisector and a circle, then shade the area that fits both rules.

Section 4: Bearings

Bearings are a way of describing direction using angles. There are three golden rules for bearings:

  1. Measured from North: Always start your protractor at the North line (\(0^{\circ}\)).
  2. Measured Clockwise: Always turn your protractor to the right.
  3. Three Digits: Always write bearings with three numbers. For example, \(45^{\circ}\) must be written as \(045^{\circ}\).

Calculating "Back Bearings"

If the bearing of \(B\) from \(A\) is \(070^{\circ}\), what is the bearing of \(A\) from \(B\)?

Because North lines are parallel, we can use interior angles (which sum to \(180^{\circ}\)).

  • If the bearing is less than \(180^{\circ}\), add \(180^{\circ}\).
  • If the bearing is more than \(180^{\circ}\), subtract \(180^{\circ}\).
  • Example: \(070^{\circ} + 180^{\circ} = 250^{\circ}\).

Did you know? Ships and planes use bearings instead of "left" or "right" because "left" changes depending on which way you are facing, but North is always the same!

Section 5: Scale Drawings and Maps

Since we can't draw a \(10\text{km}\) road on a piece of paper, we use scales. A scale might look like \(1:50,000\) or "\(1\text{cm} = 2\text{km}\)."

Step-by-Step for Scale Drawings:

  1. Check the scale provided in the question.
  2. Convert the real-life distance to the drawing distance. (Example: If \(1\text{cm} = 5\text{km}\), then \(20\text{km}\) is \(20 \div 5 = 4\text{cm}\)).
  3. Draw the line or locus accurately using your ruler.

Key Takeaway: Always read the question carefully to see if it asks for the distance "on the map" or the "actual distance" in real life!

Chapter Summary Checklist

  • Can I draw a perpendicular bisector and an angle bisector using only a compass and ruler?
  • Do I remember to leave my construction arcs visible?
  • Can I identify which locus to draw based on keywords like "equidistant"?
  • Do I always measure bearings clockwise from North using three digits?
  • Can I use a scale to convert map distances to real-life distances?

Don't worry if your compasses slip the first few times—it takes practice to get the tension right. Keep your pencil sharp and your North lines straight!