Welcome to Constructions and Loci!

Have you ever wondered how architects draw perfectly straight, symmetrical building plans without guessing, or how mobile phone companies figure out where to place masts so everyone gets good signal? The secret lies in two closely linked mathematical skills: constructions and loci.

In this chapter, you will learn how to produce accurate drawings using only a ruler, a pair of compasses, and a pencil, and how to track the paths of moving points according to specific rules. Don't worry if using a pair of compasses feels slightly awkward at first — with a few simple steps, you will master these techniques in no time!

Did you know? Ancient Greek mathematicians like Euclid did all their geometry using only an unmarked straight edge and a compass. In your GCSE exam, you are carrying on a tradition that is over \(2000\) years old!

The Golden Rule of Constructions: Never rub out your construction arcs! Examiners give method marks for seeing the faint intersecting arcs created by your compass. Keep them clear, neat, and visible.


Part 1: The Essential Toolkit & Golden Rules

Before you begin any construction question, make sure you have the right equipment ready:

• A sharp pencil (a blunt pencil makes your lines thick and inaccurate).
• A tight pair of compasses (if the hinge is loose, your radius will slip mid-drawing).
• A clear ruler with clear millimeter markings.
• A protractor (used only when asked to measure, or for angle constructions where compass methods are not specified).

Accuracy Target: In GCSE exams, measurements are usually expected to be accurate within \(\pm 2\text{ mm}\) for lengths and \(\pm 2^\circ\) for angles.


Part 2: Standard Geometric Constructions

1. The Perpendicular Bisector of a Line Segment

What does it mean?
• Bisector means dividing something exactly into two equal halves.
• Perpendicular means at a right angle (\(90^\circ\)).
So, the perpendicular bisector is a line that cuts another line segment exactly in half at \(90^\circ\).

Everyday Analogy: Imagine two houses, \(A\) and \(B\). If you want to build a straight fence that is always completely fair and equally far from both houses, that fence is the perpendicular bisector!

Step-by-Step Method to bisect line segment \(AB\):
1. Place the compass point on point \(A\).
2. Open your compass so that the width is more than halfway across the line segment \(AB\).
3. Draw a large arc going above and below the line \(AB\).
4. Without changing the compass width, move the compass point to point \(B\).
5. Draw another arc above and below the line so that it crosses (intersects) your first arc in two places.
6. Use your ruler and pencil to draw a straight line through the two intersection points.

Quick Review: If your arcs do not cross, your compass was opened less than halfway. Open it wider and try again!


2. The Angle Bisector

What does it mean?
An angle bisector is a straight line that cuts an angle into two perfectly equal smaller angles.

Step-by-Step Method to bisect an angle with vertex \(V\):
1. Place the compass point firmly on the corner of the angle (vertex \(V\)).
2. Draw an arc that cuts through both arms of the angle. Label or note the two points where the arc crosses the lines (let's call them \(P\) and \(Q\)).
3. Place your compass point on point \(P\) and draw an arc inside the angle space.
4. Keeping the compass set to the exact same width, place the point on \(Q\) and draw an intersecting arc.
5. Using a ruler, draw a straight line from the vertex \(V\) right through the point where those two arcs intersect.

Key Takeaway: If the starting angle was \(70^\circ\), your line will create two neat \(35^\circ\) angles.


3. Perpendicular from a Point to a Line

Sometimes you need to drop a line at \(90^\circ\) to a baseline from an external point \(P\) sitting above or below it.

Step-by-Step Method:
1. Put the compass point on the given point \(P\).
2. Open the compass wide enough so that when you draw an arc, it crosses the straight line in two separate places.
3. From each of those two intersection points on the line, draw an arc on the opposite side of the line to \(P\) (using the same compass width).
4. Draw a straight line connecting point \(P\) to the point where these new arcs intersect. This line meets the original baseline at exactly \(90^\circ\).


4. Constructing Standard Triangles

Case A: Side-Side-Side (SSS) Triangles
Example: Construct a triangle with side lengths \(8\text{ cm}\), \(6\text{ cm}\), and \(5\text{ cm}\).
1. Draw the longest side as your base using a ruler (e.g., base \(AB = 8\text{ cm}\)).
2. Open your compass to a width of \(6\text{ cm}\) using your ruler. Place the point at \(A\) and draw an arc above the base.
3. Open your compass to a width of \(5\text{ cm}\). Place the point at \(B\) and draw an arc crossing the first arc.
4. Label the intersection point \(C\). Draw straight lines from \(A\) to \(C\) and from \(B\) to \(C\).

Case B: Constructing an Equilateral Triangle (and a \(60^\circ\) Angle)
Since all sides are equal, draw a base of length \(x\text{ cm}\), then draw arcs of radius \(x\text{ cm}\) from both endpoints. Connecting the intersection creates an equilateral triangle where every interior angle is exactly \(60^\circ\)!

Memory Aid for Angle Combinations:
• Bisect a \(60^\circ\) angle to construct a \(30^\circ\) angle.
• Construct a perpendicular line (\(90^\circ\)) and bisect it to construct a \(45^\circ\) angle.


Part 3: Understanding Loci

What is a Locus?
A locus (plural: loci, pronounced "low-sigh") is the path traced out by a point that moves according to a specific mathematical rule. You can think of a locus as a boundary or a collection of all possible locations that satisfy a condition.

Everyday Analogy: Imagine a goat tied to a stake in the ground with a \(3\text{ m}\) rope. The maximum area the goat can graze is a circle. The edge of the circle is the locus of points exactly \(3\text{ m}\) away from the stake!


The 4 Fundamental Loci Rules

Almost every locus exam question is built from one or more of these four basic rules:

Rule 1: Fixed distance from a single point
• Condition: Points that are a fixed distance \(d\) from a point \(P\).
• Result: A circle with center \(P\) and radius \(d\).
• How to draw: Use your compass set to distance \(d\), put the point on \(P\), and draw a full circle (or arc).

Rule 2: Fixed distance from a straight line segment
• Condition: Points that are a fixed distance \(d\) from a straight line segment \(AB\).
• Result: A "stadium" or "pill" shape — two parallel lines on either side of \(AB\), joined at the ends by semicircles centered at \(A\) and \(B\).
• Common Mistake to Avoid: Drawing square box ends! The distance from an endpoint curves smoothly around the point, forming a semicircle of radius \(d\).

Rule 3: Equidistant from two points
• Condition: Points that are the exact same distance from point \(A\) as they are from point \(B\).
• Result: The perpendicular bisector of the line segment \(AB\).

Rule 4: Equidistant from two intersecting straight lines
• Condition: Points that are the exact same distance from line \(L_1\) and line \(L_2\).
• Result: The angle bisector between the two lines.


Part 4: Regions and Combined Loci Problems

In GCSE exam questions, you are often given a map or a garden plan and asked to shade a region that satisfies several conditions at the same time.

Step-by-Step Strategy for Region Questions:

1. Break it down: Read the question sentence by sentence. Identify which of the 4 basic rules applies to each condition.
2. Draw each boundary accurately:
• "Distance from point \(A\) is less than \(4\text{ cm}\)" \(\implies\) Draw a circle of radius \(4\text{ cm}\) centered at \(A\).
• "Closer to point \(B\) than point \(C\)" \(\implies\) Draw the perpendicular bisector between \(B\) and \(C\).
• "Closer to wall \(AB\) than wall \(AD\)" \(\implies\) Draw the angle bisector of angle \(BAD\).
3. Check the inequalities:
• "Less than \(d\text{ cm}\)" means inside the boundary.
• "More than \(d\text{ cm}\)" means outside the boundary.
• "Closer to \(A\) than \(B\)" means on the side of the bisector containing \(A\).
4. Shade clearly: Shade only the final region that satisfies all conditions simultaneously. Label the shaded region clearly (e.g., with an \(R\)) if asked.


Worked Example:

Question: A homeowner wants to plant a tree in their rectangular garden \(ABCD\) such that the tree is:
• More than \(3\text{ m}\) away from corner \(A\).
• Closer to side \(AB\) than side \(AD\).
• Closer to corner \(B\) than corner \(C\).
Scale: \(1\text{ cm} = 1\text{ m}\)

Solution Breakdown:
1. Condition 1: Set your compass to \(3\text{ cm}\). Put the compass point on corner \(A\) and draw an arc inside the garden. The tree must be outside this arc.
2. Condition 2: Construct the angle bisector of the corner angle at \(A\) (angle \(DAB\)). The tree must be on the side of this line closer to wall \(AB\).
3. Condition 3: Construct the perpendicular bisector of the line segment \(BC\). The tree must be on the side of this line closer to corner \(B\).
4. Final Step: Shade the overlapping region that satisfies all three conditions.


Part 5: Common Mistakes and Top Tips

• Blunt Pencils: Thick pencil lines cause inaccuracies of \(1\) to \(2\text{ mm}\), which can lose you marks. Keep your pencil sharp!
• Rubbing Out Arcs: Do not erase your construction marks. Examiners need to see them to award full credit.
• Flat Ends on Line Loci: Remember that the locus of points at a distance from a line segment has curved, semicircular ends around the endpoints, not flat, square corners.
• Compass Slippage: Ensure the screw on your compass is tight before going into the exam room so the legs don't widen or close unexpectedly while drawing arcs.
• Scale Conversions: Always check the scale on map questions (e.g., \(1\text{ cm} = 50\text{ m}\)). Convert distances carefully before setting your compass radius.


Chapter Summary Checklist

Before moving on, make sure you can confidently:
• Construct the perpendicular bisector of a line segment.
• Construct the bisector of a given angle.
• Construct a perpendicular from a point to a line and at a point on a line.
• Construct triangles given three sides (SSS) or angles using compasses.
• Draw the locus of points equidistant from a point, a line, two points, and two intersecting lines.
• Combine multiple loci rules to locate points or shade defined regions on a diagram.