Chapter Notes: Construction

Table of Contents
1. Understanding the Tools of Construction
2. Copying a Segment
3. Copying an Angle
4. Constructing Perpendicular Lines
5. Constructing Parallel Lines
6. Constructing an Equilateral Triangle
7. Bisecting an Angle
8. Why Constructions Work
9. Applications of Geometric Construction
10. Tips for Successful Construction
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Geometric construction is an ancient and powerful way of creating accurate drawings using only two simple tools: a compass and a straightedge. Unlike sketching or measuring with a ruler, construction involves following precise step-by-step procedures to produce perfect geometric figures. When you construct something in geometry, you are using mathematical reasoning and these two tools to guarantee that your figure has exact properties. This method has been used for thousands of years and is still important today in fields like architecture, engineering, and design.

Understanding the Tools of Construction

Before you can construct geometric figures, you need to understand the two basic tools you will use and what they can do.

The Compass

A compass is a tool with two arms that open and close. One arm has a sharp point that stays fixed on the paper, while the other arm has a pencil or marking tool. The compass is used to draw circles and arcs (parts of circles). The most important feature of a compass is that it can copy distances. When you set the compass to a certain width, that distance stays the same as you move the compass to different places on your paper.

Think of a compass like a robot arm that can swing around a fixed point while keeping exactly the same length. Wherever you move the center point, the pencil end traces out a perfect circle with that same radius.

The Straightedge

A straightedge is a tool with a perfectly straight edge used for drawing straight lines. It looks like a ruler, but there is an important difference: when doing constructions, you do not use the measuring marks on the ruler. You only use the straight edge itself to connect points or extend lines. This is why we call it a straightedge rather than a ruler.

Imagine you have a ruler but someone covered all the numbers with tape. You can still draw perfectly straight lines between any two points, but you cannot measure how long they are. That is how a straightedge works in construction.

Basic Rules of Construction

When performing geometric constructions, there are a few important rules to follow:

• You may only use a compass and straightedge.
• You cannot measure distances or angles directly.
• You cannot mark measurements on your straightedge.
• All distances are copied using the compass.
• Every construction must work exactly, not approximately.

Copying a Segment

One of the simplest and most fundamental constructions is copying a segment. This means creating a new line segment that has exactly the same length as a given segment, without measuring it. This construction teaches you how to use the compass to transfer distances.

Steps to Copy a Segment

Suppose you want to copy segment AB to create a new segment starting at point C.

1. Place the compass point on point A and the pencil on point B. This sets the compass width to exactly the length of segment AB.
2. Without changing the compass width, move the compass so the point is on C.
3. Draw an arc with the pencil end of the compass. Any point on this arc is exactly the same distance from C as B is from A.
4. Use your straightedge to draw a ray (a line extending in one direction) from point C.
5. Mark the point where the arc crosses the ray. Call this point D.
6. Segment CD is now exactly the same length as segment AB.

Example:  You have a line segment labeled PQ that is 4.5 cm long (though you do not know this measurement).
You need to construct a new segment starting at point R that has the exact same length as PQ.

How do you construct this segment?

Solution:

Place your compass point on P and open it until the pencil reaches Q. This captures the distance PQ.

Draw a ray starting at point R extending to the right.

Without changing the compass setting, place the compass point on R and draw an arc that crosses the ray.

Label the intersection point S. The segment RS now has exactly the same length as PQ, which means RS = PQ.

The new segment RS is congruent to segment PQ, meaning RS ≅ PQ.

Copying an Angle

Another essential construction is copying an angle. This creates a new angle that has exactly the same measure as a given angle, without using a protractor to measure degrees.

Steps to Copy an Angle

Suppose you want to copy angle BAC to create a new angle starting at point P.

1. Draw a ray from point P. This will become one side of your new angle.
2. Place the compass point on vertex A (the corner of the original angle) and draw an arc that crosses both sides of the angle. Label these crossing points D and E.
3. Without changing the compass width, place the compass point on P and draw a similar arc that crosses the ray you drew. Label the crossing point Q.
4. Now set your compass to the distance between D and E on the original angle.
5. Place the compass point on Q and draw an arc that intersects the arc you drew from P. Label this intersection point R.
6. Use your straightedge to draw a ray from P through R.
7. Angle QPR is now exactly the same measure as angle BAC.

Example:  Angle XYZ measures 65° (though you do not know this measurement).
You need to construct a new angle at point M that has the same measure as angle XYZ.

How do you construct this angle?

Solution:

Draw a ray starting at M extending to the right. This will be one side of the new angle.

Place the compass point on Y and draw an arc crossing both sides of angle XYZ. Label these points A and B.

Using the same compass width, place the compass on M and draw an arc crossing the ray. Label this point C.

Measure the distance from A to B by placing the compass point on A and the pencil on B.

Place the compass point on C and draw an arc intersecting the first arc from M. Label this intersection D.

Draw a ray from M through D. The angle formed is angle CMD ≅ angle XYZ.

The new angle at M has exactly the same measure as the original angle, which is 65°.

Constructing Perpendicular Lines

Perpendicular lines are lines that meet at a 90° angle, forming a right angle. Being able to construct perpendicular lines is crucial for many geometric applications.

Constructing a Perpendicular Bisector of a Segment

A perpendicular bisector is a line that cuts a segment exactly in half (bisects it) and forms a 90° angle with the segment. This construction creates two important things at once: it finds the midpoint of the segment and creates a perpendicular line.

Steps to Construct a Perpendicular Bisector

Given segment AB:

1. Place the compass point on A and open it to more than half the length of AB.
2. Draw an arc above and below the segment.
3. Without changing the compass width, place the compass point on B and draw arcs above and below the segment that intersect the first arcs.
4. Label the two intersection points C (above) and D (below).
5. Use your straightedge to draw line CD.
6. Line CD is the perpendicular bisector of segment AB. It crosses AB at its midpoint and forms 90° angles.

Example:  You have segment JK that is 8 cm long.
You need to find the exact midpoint of JK and draw a line perpendicular to JK at that midpoint.

How do you construct the perpendicular bisector?

Solution:

Open your compass to more than 4 cm (more than half of 8 cm), but you do not need to measure this.

Place the compass point on J and draw arcs above and below the segment.

Keep the same compass width, place the point on K, and draw arcs that cross the first arcs. Label the intersections M and N.

Draw a line through M and N using the straightedge.

The line MN crosses JK at point P, which is the midpoint. Point P is exactly 4 cm from both J and K, and line MN is perpendicular to JK.

You have successfully constructed the perpendicular bisector, and P is the midpoint of JK.

Constructing a Perpendicular from a Point to a Line

Sometimes you need to construct a perpendicular line from a point that is not on a line down to that line. This is useful for finding the shortest distance from a point to a line.

Steps to Construct a Perpendicular from a Point to a Line

Given line ℓ and point P not on the line:

1. Place the compass point on P and draw an arc that crosses line ℓ in two places. Label these points A and B.
2. Place the compass point on A and draw an arc on the opposite side of the line from P.
3. Without changing the compass width, place the compass point on B and draw another arc that intersects the first arc. Label this intersection Q.
4. Use your straightedge to draw line PQ.
5. Line PQ is perpendicular to line ℓ.

Constructing a Perpendicular at a Point on a Line

You can also construct a perpendicular line at a specific point that is already on a line.

Steps to Construct a Perpendicular at a Point on a Line

Given line ℓ and point P on the line:

1. Place the compass point on P and draw equal arcs on both sides of P along the line. Label these points A and B.
2. Open the compass wider (to more than the distance from A to P).
3. Place the compass point on A and draw an arc above (or below) the line.
4. Without changing the compass width, place the compass point on B and draw another arc that intersects the first arc. Label this intersection Q.
5. Use your straightedge to draw line PQ.
6. Line PQ is perpendicular to line ℓ at point P.

Constructing Parallel Lines

Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. They remain the same distance apart everywhere. Constructing parallel lines involves creating equal corresponding angles.

Steps to Construct a Parallel Line Through a Given Point

Given line ℓ and point P not on the line:

1. Draw a line through P that crosses line ℓ. This is called a transversal. Label the intersection point Q.
2. You now have an angle formed where the transversal crosses line ℓ. You will copy this angle at point P.
3. Place the compass point on Q and draw an arc that crosses both the transversal and line ℓ. Label these crossing points A and B.
4. Without changing the compass width, place the compass point on P and draw a similar arc crossing the transversal. Label this crossing point C.
5. Set the compass to the distance between A and B.
6. Place the compass point on C and draw an arc that intersects the arc from P. Label this intersection D.
7. Draw a line through P and D.
8. This line through P is parallel to line ℓ.

Example:  You have a horizontal line labeled ℓ and a point P located 3 cm above the line.
You need to construct a new line through P that is parallel to line ℓ.

How do you construct the parallel line?

Solution:

Draw any line through P that intersects line ℓ. Call the intersection point Q.

Place the compass on Q and draw an arc crossing both lines at the intersection. Label the points where the arc crosses as A (on ℓ) and B (on the transversal toward P).

Place the compass on P with the same width and draw an arc crossing the transversal. Label this point C.

Measure the distance from A to B by setting the compass from A to B.

Place the compass point on C and mark point D where the arc intersects the previous arc from P.

Draw a line through P and D. This line is parallel to line ℓ.

The new line through P will remain exactly 3 cm away from line ℓ at all points.

Constructing an Equilateral Triangle

An equilateral triangle is a triangle with all three sides the same length and all three angles equal to 60°. This is one of the most elegant constructions because it demonstrates the power of the compass beautifully.

Steps to Construct an Equilateral Triangle

Given segment AB as one side:

1. Place the compass point on A and set the width to reach B.
2. Draw an arc above (or below) the segment.
3. Without changing the compass width, place the compass point on B and draw another arc that intersects the first arc. Label this intersection point C.
4. Use your straightedge to draw segments AC and BC.
5. Triangle ABC is equilateral, meaning AB = AC = BC and all angles equal 60°.

Example:  You want to construct an equilateral triangle where each side measures exactly 5 cm.
You start with segment MN that is 5 cm long.

How do you construct the equilateral triangle?

Solution:

Place the compass point on M and open it until the pencil reaches N (capturing the 5 cm distance).

Draw an arc above segment MN.

Without changing the compass setting, place the compass point on N and draw another arc that crosses the first arc. Label the intersection P.

Draw segments MP and NP with the straightedge.

Triangle MNP is equilateral with all three sides equal to 5 cm and all three angles equal to 60°.

You have successfully constructed triangle MNP as an equilateral triangle.

Bisecting an Angle

To bisect an angle means to divide it into two equal parts. An angle bisector is a ray that splits an angle into two congruent angles, each with half the measure of the original angle.

Steps to Bisect an Angle

Given angle ABC (where B is the vertex):

1. Place the compass point on vertex B and draw an arc that crosses both sides of the angle. Label these crossing points D and E.
2. Place the compass point on D and draw an arc in the interior of the angle.
3. Without changing the compass width, place the compass point on E and draw another arc that intersects the first arc. Label this intersection point F.
4. Use your straightedge to draw ray BF.
5. Ray BF is the angle bisector. It divides angle ABC into two equal angles: angle ABF and angle FBC.

Example:  You have angle RST that measures 80°.
You need to construct the angle bisector to divide this angle into two equal parts.

What is the measure of each resulting angle?

Solution:

Place the compass point on vertex S and draw an arc crossing both SR and ST. Label the crossing points A and B.

Place the compass on A and draw an arc inside the angle.

With the same compass width, place the compass on B and draw an arc intersecting the previous arc. Label the intersection C.

Draw ray SC with the straightedge.

Ray SC bisects angle RST, creating two angles: angle RSC and angle CST. Each angle measures 80° ÷ 2 = 40°.

The angle bisector divides the 80° angle into two 40° angles.

Why Constructions Work

You might wonder why these constructions always produce exact results. The answer lies in the geometric properties and theorems that guarantee the relationships between the parts.

Properties That Make Constructions Reliable

• Circles preserve distance: Every point on a circle is exactly the same distance from the center. When you draw an arc with a compass, you are creating part of a circle, so all points on that arc are equidistant from the center point.
• Congruent triangles: Many constructions create congruent triangles, which have corresponding parts that are equal. The perpendicular bisector construction, for example, creates two congruent triangles.
• Properties of symmetry: Constructions often rely on symmetry. The perpendicular bisector is a line of symmetry for the segment, and points on the bisector are equidistant from both endpoints.
• Angle relationships: When constructing parallel lines, we use the fact that corresponding angles created by a transversal crossing parallel lines are equal.

The Power of Construction

Constructions prove that geometric figures can be created exactly, not just approximately. When you construct something, you know with mathematical certainty that the relationships are perfect. A constructed right angle is exactly 90°, not 89.9° or 90.1°. A constructed angle bisector divides an angle into exactly two equal parts, not approximately equal parts.

Think of constructions as recipes that always work. If you follow the steps exactly with the right tools, you will always get the perfect result, just as a recipe always produces the same dish if you follow the directions precisely.

Applications of Geometric Construction

Geometric constructions are not just theoretical exercises. They have practical applications in many fields.

Architecture and Design

Architects use construction principles when designing buildings. Finding perpendicular lines helps create walls that meet at right angles. Bisecting angles helps create symmetrical designs. Before modern computer tools, architects relied heavily on compass and straightedge constructions to create precise blueprints.

Engineering

Engineers use geometric constructions when designing machines, bridges, and structures. Parallel lines ensure that components fit together properly. Perpendicular constructions help create stable, balanced structures.

Art

Artists use geometric constructions to create precise patterns, mandalas, and designs. Islamic art, in particular, features intricate geometric patterns that were created using compass and straightedge constructions. The construction of equilateral triangles and regular polygons forms the basis of many beautiful artistic designs.

Navigation and Mapping

Historically, navigation relied on geometric constructions to plot courses and measure angles. Even today, understanding constructions helps in reading and creating maps.

Tips for Successful Construction

Mastering geometric construction takes practice, but these tips will help you succeed:

• Use sharp pencils: A sharp pencil makes precise marks. Dull pencils create thick lines that make it hard to see exact points.
• Draw lightly at first: Make light construction marks so you can see all your work. You can darken the final answer later.
• Keep your compass tight: Make sure the compass joint is tight enough that the width does not change accidentally as you draw.
• Mark points clearly: Label all important points with clear letters so you can follow the steps.
• Do not change compass width mid-construction: Many constructions require keeping the same compass width for multiple steps. Be careful not to bump the compass.
• Check your work: After completing a construction, verify that it looks correct. For example, if you constructed a perpendicular bisector, check that it appears to create right angles.
• Practice basic constructions: Master the fundamental constructions (copying segments, copying angles, bisecting) before attempting more complex figures.

Geometric construction is a beautiful blend of art and mathematics. With just a compass and straightedge, you can create perfect geometric relationships that have fascinated mathematicians for thousands of years. As you practice these constructions, you develop precision, spatial reasoning, and an appreciation for the elegant logic of geometry.