Geometric Constructions
Euclid, a Greek mathematician known as the “Father of Geometry,” wrote the book Elements, which
recorded all of the mathematical knowledge of the time in an organized and logical fashion. Since 1482,
more than a thousand editions of Elements have been published in many languages. In fact, it was
considered to be required reading by all educated people until the twentieth century, and was still being
used as a high school textbook as recently as 1980. In Elements, Euclid used construction techniques
extensively, and so they have become a part of the geometry field of study. A compass and straightedge
are used to create constructions.
As you complete the assignment, keep this question in mind:
How can congruent segments and lengths be created with a tool with no marked measurements?
In this task, you will apply what you have learned in this lesson to answer this question.
Directions
Answer each of the following questions, reading the directions carefully as you go. Refer to the
constructions you completed in the previous assignment to help answer the questions.Type all your
responses into this document so you can submit it to your teacher for a grade. You will be given partial
credit based on the completeness and accuracy of your explanations.
Your teacher will give you further directions about how to submit your work. You may be asked to upload
the document, e-mail it to your teacher, or print it and hand in a hard copy.
Now, let’s get started!
Step 1: Copy a segment and an angle.
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Which step in the construction of copying a line segment ensures that the new line segment has
the same length as the original line segment?
Copying a line segment can be done by utilizing a compass. The process involves using the
compass to measure the width of the line segment (such as if a segment is an inch in legenth
then you would expand the compass to cover one inch) and then placing the instrument onto the
point you are trying to mark over. After doing that you would then create an arc and mark the
opposite point from the original, once that is completed with a line drawn through connecting
each point you will have succesfully copied the line segment.
•
Explain how you could use the construction tool or a compass and straightedge to create a line
segment that is twice as long as .
To utilize a compass to create a line segment that is double to the length of the one you have
measured would involve first copying the line segment, thus resulting in two identical lines. Then,
place the compass onto the opposite end of the copied line and create an arc that is equal in
length to the other lines. Finally it is important to draw a line passing through the arc so that the
copied line and the expansion are connected.
•
The construction of copying is started below. The next step is to set the width of the compass to
the length of . How does this step ensure that the new angle will be congruent to the original
angle?
it ensures that the new angle will be congruent because it measures and replicates a 180 o angle
•
How is copying a line segment similar to copying an angle?
Copying a line segment and copying an angle both require the utilization of a compass or similar
instrument to measure the length of the original line segment or angle, as well as the displacement to
copy the aforementioned item.
Step 2: Construct a perpendicular line.
•
In the step shown of the construction of a line through a point that is perpendicular to the given
line, why must the compass point be placed on points A and B? How would the construction be
different if the compass point were placed at random points on the original line?
It is very important to ensure that the points of the compass are put onto points A and B because
not doing so could lead to a perpendicular line not being made. Placing the compass points at
random locations on the line would cause the perpendicular line to be offset and not properly
created.
Step 3: Construct an angle and a perpendicular bisector.
•
The construction of creating the perpendicular bisector of is started below. How would the
construction be different if you changed the compass setting in the next step of the perpendicular
bisector construction?
By changing the setting of the compass you would cause the arc to be displaced, thus resulting in
a line segment which doesn't properly create a bisector of line PQ.