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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.**

Explore

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.

How is $AB$ and the line passing through the circles' points of intersection related? There are several basic constructions in Geometry.

- 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

Discussion

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 $AB$

Using a straightedge, draw a segment longer than the original.

2

Measure $AB$ with a Compass

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

The compass is now set to the correct length. After moving the compass to the new segment, mark where the end should be with the pencil.

This mark shows where to place the endpoint of the new segment.

It has been shown that $A_{′}B_{′}$ has the same length as $AB.$

Discussion

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 $A$

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

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

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 $AB$

Any line that passes through the midpoint of $AB$ is a bisector of $AB.$

Discussion

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, these three steps can be followed.1

Draw an Arc Centered at $A$

2

Draw an Arc Centered at $B$ with the Same Radius

Keeping the same compass length, draw another arc on the opposite side. 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

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.

Example

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.

See solution.

Choose one side and draw its perpendicular bisector.

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!

Discussion

A line perpendicular to a given line through a point can be drawn by using a compass and straightedge. This perpendicular line is unique and is described by the Perpendicular Postulate.

Given a line and a point, a perpendicular line through the point can be drawn by following these steps.

1

Draw a Circle Across the Line

2

Draw Two Arcs That Intersect Each Other

Next, keeping the compass set, place its needle point at a point where the line intersects the circle. Draw an arc. Then from the other point of intersection draw another arc. The arcs should intersect each other outside the circle.

3

Use a Straightedge to Draw the Perpendicular Line

Finally, use a straightedge to draw the line through the given point and the point of intersection between the arcs. This line is perpendicular to the given line.

Explore

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 $⊙C.$

What can be concluded about the ray and the angle? Is it possible to find a segment that is perpendicular to the ray?

Discussion

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 four steps.1

Draw a Ray with Endpoint $D$

To begin, draw a ray using a straightedge.

2

Draw Arcs Across the Rays

Next, place the needle point of the compass at the vertex of the angle. Then, draw an arc across the rays at any distance from the vertex. With the same compass setting, position the needle point at $D,$ and draw an arc across the ray.

3

Draw an Arc With Radius $BC$ and Center $E$

Bring the compass back to the original angle. Adjust the compass to measure the distance between the rays at their points of intersection. With that measurement, return to the what is becoming the copy. Align the needle point of the compass onto the ray and the arc's point of intersection. Mark this distance onto the arc.

4

Draw a Ray From $D$ Through $F$

On the copy, the second ray can be drawn from the endpoint on the first ray. Draw a segment from this point through the marked position on the arc.

Discussion

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

2

Draw Two New Arcs Intersecting Each Other

Next, keeping the compass set, 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

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.

Discussion

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 line through a point $P$ exists that is parallel to a line $ℓ.$ The line through $P$ can be found using 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

Begin by drawing a line that intersects $ℓ,$ at any arbitrary location, and goes through the point. The point where the new line intersects $ℓ$ will be named $Q.$

2

Transfer a Copy of the Angle to $P$

A copy of the angle formed by $ℓ$ and $QP $ will be constructed in order to transfer the copy to point $P.$ To copy the angle, begin by fixing the needle of the compass at $Q.$ With the compass set to an arbitrary length, draw an arc on $QP $ between $Q$ and $P.$ Then, with the same setting, draw an arc from $P$ across $QP .$

Adjust the radius of the compass so that both compass ends are on the first drawn arc's intersection points with $ℓ$ and $QP .$ Using that setting, place the needle of the compass where the second arc intersects $QP .$ Draw an arc intersecting the second arc. The point where these arcs intersect can be labeled $R.$

3

Draw a Line Through $P$ and $R$

Draw the line that connects points $P$ and $R.$

This line is parallel to $ℓ.$

Example

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.

See solution.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

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.

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.

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 $AB=AD=DC.$

Closure

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! Designers might consult constructions when creating company logos. For example, the Apple logo can be constructed using a compass and a straightedge. Take a look at some other designs and think about how they were made using a construction.

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