Notes

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?

One yo draw a line with a straight edge you take the compass and make sure the center part of the compass and the pencil are touching both points of the original line segment, then you take that to the line you created and make a arc. Make sure to put a point at the starting line you created before making the arc.


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







Using the straight edge, draw a straight line that is more than twice the length of AB. Make a mark on the line at the point where you want A. Secondly set the compass width to the length of AB. Third, With the the compass at point at A mark on your line at A that is the length of AB from A. Lastly, Move the compass to point a and mark the point B on your line that is the length of AB from A (and twice the length of AB from A)















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?

The Angle QPR is equal to the angle TSU because triangle APB and triangle TSU are congruent by SSS postulate.








How is copying a line segment similar to copying an angle? We use a straight edge and compass to draw a copy of line segment or copy of an angle. To create a copy of a angle, two line segments are created by copying them from the 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? The perpendicular line goes through the given point. You would get a perpendicular line at some random location.


















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? If the compass angle is changed the entire geometric shape being drawn is different. The arcs made by it wont intersect each other at all. So the construction of the perpendicular bisector would not be possible.