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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.

## Using Circles to Construct Lines

Consider a segment with endpoints and There are two circles with the same radius — centered at and respectively. The radius can be adjusted, and a line that passes through the two circles' points of intersection can be drawn.
How is and the line passing through the circles' points of intersection related? There are several basic constructions in Geometry.

## 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 these three steps can be followed.
1
Draw a Segment Longer Than
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Using a straightedge, draw a segment longer than the original.
2
Measure with a Compass
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To make a segment with the same length as 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 on the New Segment
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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 has the same length as

## 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 will be constructed using a compass and a straightedge.
1
Draw an Arc Centered at
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Place a compass with its needle point at 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 with the Same Radius
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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
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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
4
Bisectors of
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Any line that passes through the midpoint of is a bisector of

## 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, these three steps can be followed.
1
Draw an Arc Centered at
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Begin by placing the sharp end of the compass at one of the endpoints of the segment. Then, draw an arc with a radius larger than half the length of the segment.
2
Draw an Arc Centered at with the Same Radius
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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
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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.
When each arc in this construction is completed into a circle, the graph in the exploration slide is obtained. Therefore, the line drawn in the exploration slide is the perpendicular bisector of

## 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.

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 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!

## Constructing a Perpendicular Line

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
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Begin by placing the needle point of the compass at the point. Then, draw a circle in such a way that the line is secant to the drawn circle.
2
Draw Two Arcs That Intersect Each Other
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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
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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.

## 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 The applet shows three circles on the angle. Two of the circles, and have the same radius. Examine the ray with a starting point and passing through the intersection points of and

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

## 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 four steps.
1
Draw a Ray with Endpoint
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To begin, draw a ray using a straightedge.
2
Draw Arcs Across the Rays
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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 and draw an arc across the ray.
3
Draw an Arc With Radius and Center
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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 Through
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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.

## 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
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Begin by placing the sharp end of the compass at the vertex of the angle. Draw an arc across the rays.

2
Draw Two New Arcs Intersecting Each Other
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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
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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.

## 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 line through a point exists that is parallel to a line The line through can be found using a compass and a straightedge.

The parallel line can be constructed in several steps.

1
Draw a Line Through and Across at an Angle
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Begin by drawing a line that intersects at any arbitrary location, and goes through the point. The point where the new line intersects will be named

2
Transfer a Copy of the Angle to
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A copy of the angle formed by and will be constructed in order to transfer the copy to point To copy the angle, begin by fixing the needle of the compass at With the compass set to an arbitrary length, draw an arc on between and Then, with the same setting, draw an arc from across

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

3
Draw a Line Through and
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Draw the line that connects points and
This line is parallel to

## Creating a Trapezoid by Constructing Parallel Lines

Maya and Tearrik are playing a construction game that they call Construct It if You Can! Tearrik starts the game by drawing and as shown. It is then Maya's turn to complete the drawing into a trapezoid. There is one catch to the game rules: Maya is only allowed to use copies of to complete the trapezoid.

Help Maya construct the trapezoid that Tearrik desired by using compass, and a straightedge.

See solution.

### Hint

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

### Solution

It seems that looks shorter than and Maya can only use copies of Therefore, one of the sides that will be drawn should be parallel to 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 and through there are three steps to follow.

1. Draw a Line through and across at an angle
2. Construct a copy of the angle
3. Draw a line through and the point created by the two intersecting arcs

To make the next steps easier, ought to be extended using a straightedge.

Here, a copy of will be constructed in such a way that the copy will be located at point To do so, draw an arc centered at across the angle. Then, with the same setting, draw another arc centered at

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 and draw an arc that intersects the second arc. Name the arcs' point of intersection

Use a straightedge to draw the line that connects points and
The line is parallel to Now, on this line, a segment with a length 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, and can be connected. After deleting the auxiliary arcs and lines, a trapezoid can be seen.

Note that is also equal to Use a compass to check

## 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!
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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