14. Compass and Straight Edge Constructions
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Chapter 1
14.
Compass and Straight Edge Constructions
Compass and straightedge constructions are foundational techniques that have been used for millennia. With these tools, one can create precise shapes and figures without the need for numerical measurements. One of the intriguing tasks is constructing a trapezoid using only a compass and straightedge. This technique requires a keen understanding of geometric principles and a steady hand. Such constructions not only hone one's skills but also provide a deeper appreciation for the intricacies of geometry.
| Student Learning Objectives: |
|---|
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| | 13 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Compass and Straight Edge Constructions
Slide of 13
Construction in Geometry is a process of creating accurate geometric shapes by following specific steps. Constructions can be made using various tools and methods such as a compass, a straightedge, string, reflective devices, and paper folding. In this lesson, a compass and a straightedge will be used to construct geometric shapes.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Using Circles to Construct Lines
Consider a segment with endpoints A and B. There are two circles with the same radius — centered at A and B, respectively. The radius can be adjusted, and a line that passes through the two circles' points of intersection can be drawn.
- Copying a segment
- Copying an angle
- Bisecting a Segment / Drawing a perpendicular bisector
- Bisecting an angle
- Drawing a perpendicular line through a given point
- Drawing a line parallel to a given line and through a given point
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Discussion
Constructing a Copy of a Segment
Throughout the lesson, the constructions mentioned above will be performed. It is best to start with the most straightforward construction, that is, copying a segment.
Using a straightedge and a compass, it is possible to construct a copy of a segment.
To copy AB, these three steps can be followed.
1
Draw a Segment Longer Than AB
expand_more Using a straightedge, draw a segment longer than the original. 
2
Measure AB with a Compass
expand_more To make a segment with the same length as AB, measure the original segment with a compass. To do so, place the needle point on one of its endpoints and place the pencil on the other end.
3
Mark Point B' on the New Segment
expand_more Using the same compass setting as Step 2, place the needle point on point A' and draw an arc that intersects A'C. Label the intersection B'.
Point B' corresponds to the endpoint of the new segment.
The resulting segment A'B' is congruent to AB. A'B'≅AB
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Discussion
Constructing a Bisector of a Segment
Now, it will be shown how to construct the geometric objects that bisect a segment.
To bisect a segment means to draw the object that bisects it, or passes through its midpoint.
Here, a bisector of AB will be constructed using a compass and a straightedge.
1
Draw an Arc Centered at A
expand_more Place a compass with its needle point at A. Set the compass to go beyond half the length of the segment. Then draw an arc that intersects the segment.
2
Draw an Arc Centered at B with the Same Radius
expand_more Keep the compass set to the same length. Then place the needle point at the other end of the segment and draw a second arc across the segment. 
If the arcs have been drawn large enough, they will now intersect each other twice. If they don't, extend them.
3
Draw a Line Through the Points of Intersections of Arcs
expand_more Use the straightedge to draw a line through the intersection points of the arcs. The point where the line and the segment intersect is the midpoint of AB.
4
Bisectors of AB
expand_more Any line that passes through the midpoint of AB is a bisector of AB.
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Discussion
Constructing the Perpendicular Bisector of a Segment
The steps taken to bisect a segment are also used to construct a perpendicular bisector.
On any given segment, a compass and a straightedge can be used to draw its perpendicular bisector.
To draw the perpendicular bisector of a segment AB, these three steps can be followed.
1
Draw an Arc Centered at A
expand_more 
2
Draw an Arc Centered at B with the Same Radius
expand_more Keeping the same compass length, draw another arc on B, the other endpoint of the segment. Since the measure of the compass is greater than half the length of the segment, the two arcs should intersect at two distinct points.
3
Draw the Line Through the Points of Intersections of Arcs
expand_more Using a straightedge, draw the line through the points of intersection of the arcs.
This line is perpendicular to the given segment and their intersection is at the midpoint of the segment. Therefore, it is the desired perpendicular bisector.
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Example
Solving Problems by Constructing Bisectors
Constructions, such as drawing a line, can come in handy when building things in real life. For example, in carpentry, it is common practice to divide wooden material in half.
Vincenzo likes to practice wood carving in his free time. He is given a basswood log, from his abuelita. The log has already been cut into a square shape. To make a carving he has in mind, he wants to reshape the log into a rectangular shape with a height that is twice its width.
Help Vincenzo draw a line that divides the log into a rectangle and meets the given characteristics. He would then be able to use this line to cut the wood to obtain the shape he wants.
Answer
See solution.
Hint
Choose one side and draw its perpendicular bisector.
Solution
The perpendicular bisector of one side will divide the log in half, creating two rectangles. 
To draw the perpendicular bisector, place the tip of the compass on one of the vertices of the square log. Just by eyeballing, open the compass beyond the midpoint of AB, and draw an arc. Afterwards, place the tip on the other adjacent endpoint and draw another arc with the same radius.
Repeat this procedure using a larger width than the previous one.
Finally, using a straightedge, draw the line that connects the intersection points of the arcs.
As can be seen, with the help of a straightedge and a compass, the perpendicular bisector has been drawn. By cutting the wood with the help of this line, Vincenzo can get the rectangular piece with a height that is twice its width. He even ends up with a second piece!
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Discussion
Constructing a Perpendicular Line
The Perpendicular Postulate states that for a given line and a point not on the line, there is exactly one line through the point that is perpendicular to the given line. We can construct the perpendicular line through P using a compass and a straightedge.
1
Draw a Circle Across the Line
expand_more We begin by placing the needle point of the compass at P. We then draw a circle with a radius large enough that it intersects the line at two points.
2
Draw Two Arcs That Intersect Each Other
expand_more Next, keeping the compass set, we place the needle point at one of the points where the line intersects the circle and draw an arc. We then place the needle point at the other point of intersection and draw another arc. The two arcs should intersect outside the circle.
3
Use a Straightedge to Draw the Perpendicular Line
expand_more Finally, we use a straightedge to draw the line through the given point and the intersection of the two arcs. This line is perpendicular to the given line.
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Explore
Using Circles to Construct Angles
In the second half of the lesson, the other basic constructions that involve angles will be understood. Consider the angle BAC. The applet shows three circles on the angle. Two of the circles, ⊙ B and ⊙ C, have the same radius. Examine the ray with a starting point A and passing through the intersection points of ⊙ B and ⊙ C.
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Discussion
Constructing a Copy of an Angle
An angle can be constructed as a copy of a given angle using a compass and a straightedge.
The given angle can be copied in five steps.
1
Draw a New Ray
expand_more To begin, draw a ray using a straightedge. 
2
Draw an Arc in the Original Angle
expand_more Next, place the needle point of the compass at the vertex of the angle, A. Draw an arc across the rays at any distance from the vertex. Label B and C the intersection of the arc with each ray. 
3
Draw a an Arc in the New Ray
expand_more With the same compass setting as Step 2, position the needle point at D and draw an arc across the ray. Label E the intersection of this arc and the ray.
4
Draw an Arc With Radius BC and Center E
expand_more Adjust the compass to measure the distance between B and C. With that setting, place the needle point of the compass on E and draw a new arc that intersects the arc from Step 3. Label this intersection as F. 
5
Draw a Ray From D Through F
expand_more Using a straightedge, draw a ray from D through F. 
The resulting angle is congruent to the original angle. ∠ D ≅ ∠ A
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Discussion
Constructing the Bisector of an Angle
An angle bisector consists of points that are equidistant from the two sides of the angle, creating a ray. The applet on the previous explore slide illustrates those points and the ray.
An angle can be bisected using a compass and a straightedge.
To bisect an angle, these three steps can be followed.
1
Draw an Arc Across the Rays
expand_more 
2
Draw Two New Arcs Intersecting Each Other
expand_more Next, keeping the same compass setting, place the needle at the points where the rays intersect the arc. Draw two new arcs intersecting each other.
3
Use a Straightedge to Draw the Angle Bisector
expand_more Use a straightedge to draw a ray from the vertex through the intersection between the two arcs.
This ray is the angle bisector. It divides the angle into two congruent angles.
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Discussion
Constructing Parallel Lines
One of the great things about constructions is that some constructions can even be used to produce other constructions.
The Parallel Postulate states that exactly one parallel line to l passes through point P, and it can be constructed with a compass and a straightedge.
The parallel line can be constructed in several steps.
1
Draw a Line Through P and Across l at an Angle
expand_more 
2
Transfer a Copy of the Angle to P
expand_more Place the compass at Q and draw an arc that intersects QP. Without changing the width, draw a matching arc centered at P to copy the angle between l and QP with vertex at P.
Set the compass to the distance between the first arc's intersections. From the second arc's intersection with QP, draw an arc to meet the second arc. Label the intersection R.
3
Draw a Line Through P and R
expand_more Draw the line that connects points P and R.
The angle at Q and the angle at P are congruent corresponding angles. By the Converse Corresponding Angles Theorem, the line PR is parallel to l.
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Example
Creating a Trapezoid by Constructing Parallel Lines
Maya and Tearrik are playing a game called Construct It if You Can! Tearrik starts the game by drawing AB and BC as shown. It is then Maya's turn to complete the drawing into a trapezoid. The trapezoid can be constructed by using segments of the same length as AB. Therefore, there is one catch: Maya is only allowed to use copies of AB to complete the trapezoid.
Help Maya construct the trapezoid that Tearrik desired by using compass, and a straightedge.
Answer
See solution.
Hint
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Solution
It seems that AB looks shorter than BC and Maya can only use copies of AB. Therefore, one of the sides that will be drawn should be parallel to BC. Otherwise, the segments will not connect in the shape of a trapezoid. After the parallel line is drawn, the fourth side of the trapezoid can be drawn.
Drawing a Parallel Segment
To draw a segment parallel to BC and through A, there are three steps to follow.
- Draw a Line through A and across BC at an angle.
- Construct a copy of the angle.
- Draw a line through A and the point created by the two intersecting arcs.
To make the next steps easier, AB ought to be extended using a straightedge.
Here, a copy of ∠ B will be constructed in such a way that the copy will be located at point A. To do so, draw an arc centered at B across the angle. Then, with the same setting, draw another arc centered at A.
Adjust the compass so that both ends align with points of intersection of the first arc in relation to the two lines. With that compass setting, place the compass needle where the second arc intersects AB and draw an arc that intersects the second arc. Name the arcs' point of intersection R.
Use a straightedge to draw the line that connects points A and R.
The line AR is parallel to BC. Now, on this line, a segment with a length AB can be drawn. Measure the original segment with a compass. Then, move the compass to the new segment and mark where the end should be with a pencil.
Completing the Trapezoid
Finally, with the help of a straightedge, D and C can be connected. After deleting the auxiliary arcs and lines, a trapezoid can be seen.
Finally, use a compass to check that AB = AD = DC.
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Closure
Constructions and Real Life Applications
Considering the constructions studied throughout the lesson, it is noteworthy that the compass is used to set equidistant points and the straightedge is used to set collinearity.
Geometric constructions are practical not only for showing geometric relationships or suggesting new ones but also for making designs. For example, a flower with six petals can be drawn by a compass alone. Give it a try!
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Compass and Straight Edge Constructions
Exercise 3.1
Izabella is given the following segment.
She creates the figure below, where the semicircles are congruent, using a straightedge and a compass on a different piece of paper.
Which constructions would Isabella have used when creating the figure? Select all that apply. I. & Construction of a copy of a segment [0.5em] II. & Construction of a bisection of an angle [0.5em] III. & Construction of a perpendicular bisector [0.5em] IV. & Construction of a perpendicular line & through a given point [0.5em] V. & Construction of parallel lines
{"type":"multichoice","form":{"alts":["I","II","III","IV","V"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,2]}
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Let's take a look at the figure and see what information we can gather. Then we will draw the figure ourselves using that information.
Examining the Figure
We know that the semicircles all lie on the segment and are congruent. This means that the centers between adjacent semicircles are exactly one radius apart.
Since the semicircles are congruent, the radius of one semicircle is a quarter of the segment's length.
Drawing the Figure
Let's try to draw this figure ourselves. We can follow these four steps.
- Copy the segment.
- Locate the midpoint of A'B'.
- Locate the midpoints of A'C and CB'.
- Draw the semicircles.
The first thing we want to do is to construct a copy of AB on a different piece of paper using a compass and a straightedge.
Next we need to locate the midpoint of A'B'. We can do this by drawing the perpendicular bisector of the segment using the compass and straightedge. There are two steps to finding a perpendicular bisector.
Now we have found the midpoint of A'B', point C.
If we repeat this for A'C and CB', we can find their midpoints D and E. Notice that this also divides A'B' into quarter segments.
The three vertical lines are all perpendicular to A'B' by construction. We can conclude that these lines are parallel by the Converse Alternate Interior Angles Theorem.
The fact that the perpendicular bisectors are parallel is a result of the nature of the bisectors — that is, we did not construct them as parallel lines. This means we can disregard option V. Finally, let's construct the semicircles by using the compass. Open the compass to A'D and draw the first semicircle. Repeat this process for the remaining semicircles to finish the construction.
To draw the figure Izabella created, we copied the given segment and drew perpendicular bisectors. Therefore, the answer is I and III.
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Compass and Straight Edge Constructions
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