Ruler and compass constructions

Welcome to Ruler and Compass Constructions!

In this chapter, we are going to learn how to draw geometric shapes with incredible accuracy. Think of yourself as an architect or a designer from the days before computers! Instead of guessing where the middle of a line is, we use two simple tools—a ruler and a compass—to find it perfectly every time.

Don’t worry if this seems a bit "fiddly" at first. With a sharp pencil and a little bit of practice, you’ll be creating perfect bisectors and loci in no time!

The Geometric Toolbox

Before we start "building," let's look at our equipment (Syllabus Ref: 8.01f):

• The Ruler: We use this to draw straight lines and measure distances. In "pure" construction, we sometimes call it a "straightedge."
• The Compasses: These are for drawing circles and arcs (bits of circles). They are the most important tool for finding points that are the same distance from each other.
• The Protractor: We use this to measure and draw angles. While many constructions don't need it, it’s great for checking your work!

Pro Tip: Always keep your pencil lead sharp and make sure the screw on your compass is tight so it doesn't slip while you are drawing!

1. The Perpendicular Bisector

Syllabus Ref: 8.02a

The word perpendicular means at a right angle (\( 90^\circ \)). The word bisector means to cut something exactly in half. So, a perpendicular bisector is a line that cuts another line in half at exactly \( 90^\circ \).

Why is it useful? It helps you find the midpoint of a line segment and identifies all the points that are equidistant (the same distance) from two ends of a line.

How to construct it (Step-by-Step):

Imagine you have a line segment \( AB \):
1. Place the compass point on point \( A \).
2. Open the compass so it is more than half the length of the line.
3. Draw a large arc (a curve) that goes above and below the line.
4. Without changing the compass width, move the point to point \( B \) and draw another arc.
5. Your two arcs will cross at two points (one above the line, one below).
6. Use your ruler to draw a straight line through these two crossing points.

Common Mistake to Avoid: Changing the width of your compass between step 3 and step 4. If the width changes, your line won't be in the middle!

Key Takeaway: Every point on this new line is exactly the same distance from point \( A \) as it is from point \( B \).

2. The Angle Bisector

Syllabus Ref: 8.02b

An angle bisector is a line that cuts an angle exactly in half. If you have a \( 60^\circ \) angle, the bisector creates two \( 30^\circ \) angles.

How to construct it (Step-by-Step):

1. Place the compass point on the vertex (the corner) of the angle.
2. Draw an arc that crosses both lines (arms) of the angle.
3. Place the compass point where the arc crosses the first line and draw a small arc in the middle of the angle.
4. Keeping the same width, place the compass point where the first arc crosses the second line and draw another small arc that crosses the one you just made.
5. Use your ruler to join the vertex of the angle to the point where the two small arcs cross.

Analogy: Think of this like slicing a piece of pizza perfectly in half so you and a friend get the exact same size slice!

Key Takeaway: Any point on the bisector line is the same distance from the two lines that make the angle.

3. Perpendiculars from a Point

Syllabus Ref: 8.02c

Sometimes you need to draw a line at \( 90^\circ \) to an existing line, starting from a specific point. There are two versions of this:

A. Perpendicular at a point ON the line

1. Put the compass point on the given point.
2. Draw two arcs on the line, one on each side of the point (keeping the width the same).
3. Now, treat these two new marks like the ends of a line and follow the Perpendicular Bisector steps above!

B. Perpendicular from a point OFF the line

1. Place the compass point on the point that is "floating" above or below the line.
2. Open the compass wide enough to draw an arc that crosses the line in two places.
3. From those two crossing points, draw two arcs that cross each other on the other side of the line.
4. Use your ruler to join the original point to where the new arcs cross.

Did you know? The perpendicular distance from a point to a line is always the shortest distance. If you want to get to a wall as quickly as possible, walk at a \( 90^\circ \) angle to it!

4. Loci and Equidistant Points

Syllabus Ref: 8.02d

A Locus (plural: Loci) is a set of points that follow a specific rule. Think of it as the "path" a moving object takes if it has to follow a rule.

Common Loci Rules:

• Rule: "Exactly \( 5 cm \) from point \( X \)"
The Locus is: A circle with a radius of \( 5 cm \) and the center at \( X \).
• Rule: "Exactly the same distance (equidistant) from point \( A \) and point \( B \)"
The Locus is: The perpendicular bisector of the line \( AB \).
• Rule: "Exactly the same distance from line \( L1 \) and line \( L2 \)"
The Locus is: The angle bisector between the two lines.
• Rule: "Exactly \( 2 cm \) from a straight line"
The Locus is: Two parallel lines (one on each side) with semi-circles at the ends. We call this a "sausage shape" or a "stadium."

Quick Review Box: Key Term - Equidistant
Equi = Equal
Distant = Distance
If a point is equidistant from two things, it is exactly halfway between them.

Summary and Key Takeaways

• Constructions must be done using only a ruler and a compass. Do not rub out your construction arcs—the examiner needs to see them to give you marks!
• Use a Perpendicular Bisector to find the middle of a line or points equidistant from two points.
• Use an Angle Bisector to cut an angle in half or find points equidistant from two lines.
• A Locus is just a path or a region that follows a rule.
• The shortest distance from a point to a line is the perpendicular line.

Encouraging Phrase: If your arcs don't cross the first time, just make them a bit longer! You're doing great.