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.

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?

Explain how you could use the construction tool or a compass and straightedge to create a line segment that is twice as long as ("AB" ).

The construction of copying "QPR" is started below. The next step is to set the width of the compass to the length of ("AB" ). How does this step ensure that the new angle will be congruent to the original angle?

How is copying a line segment similar to copying an angle?

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?

Step 3: Construct an angle and a perpendicular bisector.

The construction of creating the perpendicular bisector of ("PQ" ) is started below. How would the construction be different if you changed the compass setting in the next step of the perpendicular bisector construction?