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Congruence, Proof, and Constructions

Constructions Using Different Tools

In the previous lesson, a compass and straightedge were used to make geometric constructions. In the absence of these tools, however, are other tools or methods viable options? This lesson will focus on different methods such as using a string and paper folding to make geometric constructions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Investigating a Segment's Length With a String

Ignacio needs to determine if the following segments are congruent to each other. However, he only has a piece of string and a pencil. Is he able to do it? If so, how?
Applet to measure some segments using a string and a pencil

Extra

How to Use the Applet
  • The string can be moved by clicking and dragging either endpoint.
  • To fix an endpoint at its current position, press the corresponding button.
  • To make a mark on the string, first press the Show Pencil button. Then, move the pencil tip over the place to mark. Finally, release the click.

Challenge

Locating a Triangle's Incenter Without Tools

A natural history museum manager hopes to place a replica of a moai statue in a triangular room in such a manner that the statue is equidistant from the walls that will have detailed diagrams and explanations. The location of the statue corresponds to the incenter of the triangle formed by the walls. This placement will allow for a clear walking path.

Triangle ABC
The manager's son, Kevin, sees the blueprint and is excited to help to locate the incenter. Unfortunately, he does not have a protractor. Can he still locate the incenter on the blueprint? If so, how?

Discussion

Copying a Segment With a String

When copying a segment using a straightedge and compass, the function of the compass is to measure the length of the segment. With this in mind, a string can be a good substitute for the compass.

Using a straightedge and a string, it is possible to construct a copy of a segment.

Segment AB

The following three steps can be used to draw a segment with the same length as

1

Draw a Segment Longer Than

With the straightedge, draw a new segment such that it is longer than

Drawing a segment longer than AB

2

Measure with a String
Place one end of the string at and hold it down. Once done, stretch the string so that it forms a straight line and is aligned with Then, using the pencil, mark the point on the string that is above This will be the length of
Measuring the segment AB with the string

3

Draw on the New Segment
Move the end of the string from to Once again, stretch the string so that it forms a straight line and align it with
Moving the string to the longer segment
The marked point on the string indicates the location of Then, mark this point on and label it as
Marking the point B on the longer segment
Finally, by erasing the unnecessary part of a segment whose length is the same as the length of will be obtained.
Segment A'B'

As can be seen, and are copies of each other.

Notice that the technique used in the second step of the previous construction can be used to compare the length of two segments and thereby determine if they are congruent or not.

Discussion

Copying an Angle Using a String

Using a straightedge and a string, it is possible to construct a copy of an angle.

Angle ABC

The following five steps can be used to draw an angle whose measure is equal to the measure of

1

Draw a Ray with Starting Point

To begin, draw a ray using a straightedge and label its starting point as

Angle ABC and Ray with starting point D

2

Draw Arcs Centered at and Using the String

Start by tying a pencil to one end of the string and locating the other end at The string must be shorter than the sides of the angle.

String with a pencil tied
Next, holding down the end that is on stretch the string. Keeping the string stretched, make an arc that intersects the sides of Let and be these points of intersection.
Stretching the string and making an arc
Finally, move the free end of the string to and draw an arc centered at Let be the intersection point between the ray and the arc.
Stretching the string and making an arc

3

Measure the Distance from to
To measure the distance between and untie the pencil. Then, place one end of the string at and hold it down. Next, stretch the string so that it forms a straight line that passes through
Measuring the distance from P to Q
Then, mark the point on the string that is on and tie the pencil to the string so that its tip is over the mark. It is not a necessity, but the remaining part of the string can be removed.
Marking the string at Q

4

Draw an Arc Centered on with Radius
Move the free end of the string to and make an arc that intersects the previous arc. Let be the intersection point.
Making arc at D

5

Draw

Using the straightedge, draw This ray along with form whose measure is equal to the measure of

Making arc at D

Discussion

Bisecting an Angle by Paper Folding

Paper folding, also referred to as origami, provides an alternative method for making constructions when tools such as a compass are not on hand.

An angle can be bisected using only a straightedge and folding the paper.

Angle ABC

To bisect the following two steps can be done. If available, it is recommended to use tracing paper.

1

Fold the Paper
Fold the paper in such a way that lies on Then, unfold the paper. Notice that is in the crease.
Fold the paper

2

Draw the Angle Bisector
Using a straightedge, and starting at draw the line that the crease made in the paper. This line is the angle bisector of
Draw angle bisector

Example

Solving Problems by Paper Folding

Using the folding paper technique, the challenge presented at the beginning of the lesson can be solved. A natural history museum manager hopes to place a replica of a moai statue in a triangular room in such a manner that the statue is equidistant from the walls that will have detailed diagrams and explanations. The location of the statue corresponds to the incenter of the triangle formed by the walls. This placement will allow for a clear walking path.

Triangle ABC

The manager's son, Kevin, sees the blueprint and wants to help to locate the incenter. Unfortunately, he does not have a protractor. Can he locate the incenter on the blueprint? If so, how?

Answer

Can he locate the incenter? Yes.
How? By folding the blueprint, two different angle bisectors can be drawn. The point where the angle bisectors intersect corresponds to the incenter of the triangle.
Graph:

Triangle ABC

Hint

Draw two different angle bisectors by folding the blueprint. The incenter is the point of intersection of the angle bisectors.

Solution

Kevin needs to find the incenter of the triangle formed by the room's walls.

Incenter

The incenter of a triangle is the point of intersection of the triangle's angle bisectors.

To obtain the point of intersection, Kevin needs to draw at least two angle bisectors to locate the incenter. To do so, begin by labeling the vertices of the triangle.

Triangle ABC
To draw the angle bisector of fold the blueprint so that lies on Once the blueprint is unfolded, the crease made will represent the angle bisector.
Folding the Blueprint
To draw the angle bisector the blueprint has to be folded so that lies on That way, a second crease will be made in the blueprint, representing the angle bisector of
Folding the Blueprint
The point where the two creases intersect each other corresponds to the incenter of the triangle. Consequently, this is the location where the moai replica has to be placed.
Incenter of ABC

Notice that, thanks to the Incenter Theorem, the angle bisector of does not have to be drawn to locate the incenter.

Discussion

Drawing a Perpendicular Bisector by Paper Folding

The usual way to draw the perpendicular bisector of a segment involves a straightedge and a compass. However, the same construction can be done without a compass.

Given a segment drawn on a paper, its perpendicular bisector can be drawn only using a straightedge and a pencil.

Segment AB

To draw the perpendicular bisector of the following two steps can be used. If available, tracing paper is recommended.

1

Fold the Paper
Fold the paper in such a way that matches Then, unfold the paper.
Folding Segment AB

2

Draw the Perpendicular Bisector
Using a straightedge, draw the line that the crease made in the paper. This line is the perpendicular bisector of
Drawing the perp. bisector
Notice that these two steps can be used to find the midpoint of a segment.

Example

Solving Problems by Paper Folding

Math teacher by day, the world's greatest futsal coach by night, Coach Tiffaniqua drew the following formation on a board to teach her players where to position themselves when taking a free kick near the court's center.

Futsal Field with three player forming a triangle

Ali, ecstatic, asked where he should be in the formation. Coach Tiffaniqua told Ali that he should be positioned the same distance from Diego, Ignacio, and Kevin. Determine where Ali should be positioned on the futsal field.

Answer

Diagram:
Formation including Ali

Hint

Use the fact that the circumcenter — the point of intersection of the triangle's perpendicular bisectors — is equidistant from the vertices.

Solution

Notice that Diego, Ignacio, and Kevin form a triangle. Thus, Ali should be positioned at a point that is equidistant from the vertices of this triangle.

Triangle formed by Diego, Ignacio, and Kevin
By definition, the circumcenter of a triangle is equidistant from the vertices. Therefore, to know where Ali should be placed, find the circumcenter of To do so, begin by drawing this triangle on a piece of paper.
Drawing triangle KDI on a paper
Since the circumcenter is the point of intersection of the triangle's perpendicular bisectors, at least two perpendicular bisectors need to be drawn. Start by drawing the perpendicular bisector of To do it, fold the paper so that matches
Folding the Paper
Then, draw the perpendicular bisector of by folding the paper so that and match.
Folding the Paper
The circumcenter of is the point where the two creases intersect each other. Finally, place the paper over the board and mark the circumcenter on it. This is the position where Ali should be positioned.
Marking Ali's position

Discussion

Drawing a Perpendicular Line by Paper Folding

The last construction of this lesson will be to draw a line perpendicular to a given line through a point that is not on the line. Of course, this can be done using a straightedge and compass. However, it is possible to do so without the use of a compass.

A line perpendicular to a given line through a point not on the line can be drawn by using only a straightedge and a pencil.

Line AB and point P not on the line

To draw a line perpendicular to through the following two steps can be used.

1

Fold the Paper
Fold the paper so that is on the fold, and the lines lie on each other.
Folding the Paper

2

Draw the Perpendicular Line
Using a straightedge, draw the line that the crease made in the paper. This line is perpendicular to
Drawing the Perpendicular Line

Closure

Drawing an Orthic Triangle

Ali has a piece of paper with an acute triangle drawn on it. He wants to draw the orthic triangle corresponding to whose vertices are the feet of the altitudes of

Triangle ABC

However, Ali does not have a compass at hand. Help Ali draw the desired triangle.

Answer

Diagram:

Triangle ABC

Hint

To draw the altitude from fold the paper so that is on the crease and the part of in one part of the paper falls above the other part of

Solution

To draw the orthic triangle, the three altitudes of are needed. To start, draw the altitude from by folding the paper so that is on the crease and is lined up in both parts of the paper.
Folding Triangle ABC
Using the same method, the altitudes from and can also be drawn.
Folding Triangle ABC
This construction gives the feet of the altitudes and the orthocenter of Finally, draw the orthic triangle by connecting and
Triangle ABC

Before finishing the lesson, two interesting facts about the orthic triangle — also called the pedal triangle.

  • The orthocenter of is the incenter of the orthic triangle.
  • The orthic triangle is the triangle with smallest perimeter that can be inscribed in an acute triangle.
Triangle ABC
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