📐 Chapter 5: Construction – Precision Geometry

Hello Future Engineers and Architects!

Welcome to the world of Construction! Don't worry, we aren't building houses yet; we are building perfect lines, angles, and shapes using just two simple tools: a compass and a straight edge. This chapter is all about accuracy and following specific steps (or algorithms). If you can follow a recipe, you can master construction!

Why is this important? Before computers, all design relied on these methods. Understanding construction methods ensures your geometric understanding is deep, and it guarantees you earn valuable marks for precision in the exam!


Section 1: The Essential Tools and Rules

You only need two things for classic geometric construction:

  1. A Compass: Used to draw circles and arcs, ensuring equal distances.
  2. A Straight Edge (Ruler): Used only to draw straight lines. Crucially, when constructing, you should not use the ruler's measurements unless specifically asked to draw a line of a specific starting length.
⚠️ Quick Rule Check for Exams

In construction questions, if you use a protractor (for angles) or a ruler (for precise measuring) to draw the construction, you will usually lose marks. You must show your construction arcs clearly!

🔑 Memory Tip: Think of your compass as your measuring tape and your straight edge as your guide stick. Keep the pencil sharp!


Section 2: The Big Three Fundamental Constructions

These three constructions are the foundation of almost everything else you will do. Master these, and you are halfway there!

1. Constructing the Perpendicular Bisector

The Perpendicular Bisector does two jobs at once:

  1. It cuts a line segment (AB) exactly in half (Bisector).
  2. It meets the line segment at a 90° angle (Perpendicular).

Step-by-Step Guide (Line AB):

  1. Place the compass point on end A.
  2. Adjust the compass width so it is more than half the length of AB. (If it's less than half, the arcs won't cross!)
  3. Draw a large arc above and below the line AB.
  4. Crucially: Keeping the compass width exactly the same, move the compass point to end B.
  5. Draw a second arc above and below the line AB, ensuring it crosses the first arcs.
  6. Use your straight edge to draw a line connecting the two points where the arcs intersect.

Did you know? Every single point on the perpendicular bisector is exactly the same distance (equidistant) from point A and point B.

Quick Review: Perpendicular Bisector
Job: Splits line in half and creates \( 90^\circ \).
Key Action: Compass width must be \( > \frac{1}{2} \) the line length.
2. Constructing the Angle Bisector

The Angle Bisector splits any given angle precisely into two equal smaller angles.

Step-by-Step Guide (Angle V):

  1. Place the compass point on the vertex (V) (the corner point of the angle).
  2. Draw an arc that crosses both lines of the angle. Label these intersection points P and Q.
  3. Place the compass point on P. Draw a small arc inside the angle.
  4. Keeping the compass width the same, place the compass point on Q. Draw a second small arc inside the angle, ensuring it crosses the arc from step 3.
  5. Use your straight edge to draw a line from the vertex V through the point where the two small arcs cross.

Analogy: Think of the vertex V as the starting line. You take an equal step (P and Q), and then from those equal steps, you race equally towards the middle point to find the perfect dividing line!

3. Constructing Standard Angles (\( 60^\circ \) and \( 90^\circ \))

You can construct any angle if it can be derived from the basic \( 60^\circ \) and \( 90^\circ \) constructions (like \( 30^\circ \), \( 45^\circ \), \( 120^\circ \), etc.).

A) Constructing a \( 60^\circ \) Angle

This construction is the basis for an equilateral triangle (all sides and angles equal to \( 60^\circ \)).

  1. Start with a line segment (or ray) starting at point O.
  2. Place the compass point at O. Draw a substantial arc that crosses the line (call the intersection A).
  3. Crucially: Keep the compass width exactly the same.
  4. Place the compass point at A. Draw a new arc that crosses the first arc. Call this intersection B.
  5. Draw a straight line from O through B. The angle AOB is exactly \( 60^\circ \).
B) Constructing a \( 90^\circ \) Angle (Perpendicular from a point on a line)

This is slightly different from the Perpendicular Bisector because you are creating a right angle at a specific point (P) on a line, not necessarily the middle of the line.

  1. Start with a line and mark the specific point P where you want the \( 90^\circ \) angle.
  2. Place the compass point on P. Draw arcs equally spaced on both sides of P along the line. Call these points C and D.
  3. Now, treat CD as a line segment and construct its perpendicular bisector (as you learned in point 1). Place the compass point on C, set width > PC, draw arc.
  4. Repeat from D with the same width, crossing the arc.
  5. Join P to the top intersection point. This line forms a \( 90^\circ \) angle with the original line.

Section 3: Constructing Triangles

In your exams, you might be asked to construct a triangle given specific information. You need to know the three main rules:

1. SSS (Side, Side, Side)

Given all three side lengths.

  1. Draw the longest side first using your straight edge and measurement (e.g., 8 cm). Label the ends A and B.
  2. Take your compass and set its width to the second side length (e.g., 5 cm). Place the point at A and draw a large arc.
  3. Set the compass width to the third side length (e.g., 6 cm). Place the point at B and draw an arc that crosses the first arc.
  4. The intersection point is the third vertex (C). Join AC and BC with your straight edge.
2. SAS (Side, Angle, Side)

Given two side lengths and the included angle (the angle between them).

  1. Draw the first side (e.g., 7 cm). Label the ends A and B.
  2. At the required vertex (e.g., A), construct the given angle (e.g., if it's \( 60^\circ \), use the \( 60^\circ \) construction method).
  3. Use your ruler or compass to measure the length of the second side along the newly constructed angle line (e.g., 4 cm). Mark this point C.
  4. Join B to C.
3. ASA (Angle, Side, Angle)

Given two angles and the included side (the side between them).

  1. Draw the included side (e.g., 10 cm). Label the ends A and B.
  2. At end A, construct the first angle (e.g., \( 45^\circ \)). (Hint: Construct \( 90^\circ \) and then bisect it!)
  3. At end B, construct the second angle (e.g., \( 60^\circ \)).
  4. The point where the two angled lines cross is the third vertex (C).

🚧 Common Mistake: Forgetting to show the construction arcs when asked to construct an angle like \( 60^\circ \). If you measure it with a protractor, you lose the construction marks!


Section 4: Loci and Defining Regions

The concept of Locus (plural: Loci) sounds complicated, but it just means the path or region traced out by a moving point following a specific rule.

Analogy: Imagine a dog on a leash tied to a fixed point. The fence it can walk along is the locus, and the area it can reach is the region.

Loci Rules and Their Related Constructions

Loci questions usually require you to use one of your fundamental constructions to define a boundary.

Locus 1: Equidistant from a Single Point

Rule: The locus of points exactly x cm away from a single point P.

Construction: A circle with center P and radius x.

Locus 2: Equidistant from a Line Segment (or line)

Rule: The locus of points exactly x cm away from a line AB.

Construction: Two parallel lines drawn x cm away from AB, capped by two semi-circles of radius x centered at A and B.

Locus 3: Equidistant from Two Points

Rule: The locus of points equidistant from two fixed points A and B.

Construction: The Perpendicular Bisector of the line segment AB.

🔑 Key Takeaway: If a question asks for points "closer to A than B," you must draw the perpendicular bisector and shade the side that contains A.

Locus 4: Equidistant from Two Intersecting Lines

Rule: The locus of points equidistant from two intersecting lines.

Construction: The Angle Bisector of the angle formed by those two lines.

🔑 Key Takeaway: If a question asks for points "closer to line X than line Y," you must draw the angle bisector and shade the side closer to X.

Shading Regions (Loci Inequalities)

When asked to shade the region that satisfies two or more conditions, you must:

  1. Construct all required boundaries (bisectors, circles, parallel lines).
  2. Determine which side of the boundary is the "allowed" area.
  3. Shade the area that overlaps and satisfies all conditions simultaneously.

Example Conditions:

  • Closer to A than B: Shade the side of the perpendicular bisector containing A.
  • Less than 5 cm from point C: Shade the inside of the circle of radius 5 cm centered at C.
  • More than 3 cm from line L: Shade the area outside the parallel lines 3 cm away from L.

✨ Final Study Checkpoint

Construction is about mastering the technique. Practice each of the "Big Three" constructions without looking at your notes until you can do them perfectly. Remember:

  • Accuracy matters. Sharp pencil, careful compass settings!
  • Never rub out your construction arcs! They are proof of your work.
  • The Perpendicular Bisector gives you \( 90^\circ \) and divides the line.
  • The Angle Bisector divides the angle.

Don't worry if this seems tricky at first—like riding a bike, it becomes second nature with practice! You’ve got this!