§5.1 Constructions

The student will learn about:

basic Euclidean constructions.

1

Constructions

2

The word construction in geometry has a very specific meaning: the drawing of geometric items such as lines and circles using only a compass and straightedge or ruler. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

Compass

The compass is a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it.

(There are several kinds of compass – Euclidean, modern, rusty…. It has been proven that the type you use does not matter.)

3

Straight Edge

A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

4

Why we learn about constructions

The ancient Greek mathematician Euclid is the acknowledged inventor of geometry. He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. They also provide a greater insight into geometric concepts and give us tools to draw things when direct measurement is not appropriate.

5

Copy a Segment

Modern method with a modern compass.

6

Euclidean method with a Euclidean compass.

http://www.mathopenref.com/constcopysegment.html

Copy an Angle

http://www.mathopenref.com/constcopyangle.html

7

Bisect a Segment

URL:

http://www.mathsisfun.com/geometry/construct-linebisect.html

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Bisect an Angle

URL:

http://www.mathsisfun.com/geometry/construct-anglebisect.html

9

Perpendicular to a Point on the Line

URL:

http://www.mathsisfun.com/geometry/construct-perponline.html

10

Perpendicular to a Line from a Point not on the Line

URL:

http://www.mathsisfun.com/geometry/construct-perpnotline.html

11

Parallel to a Line from a Point not on the Line

URL:

http://www.mathsisfun.com/geometry/construct-paranotline.html

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Sketchpad

13

URL:

Sketchpad on Campus

BASIC CONSTRUCTIONS1. Copy a segment

2. Copy an angle.

3. Bisect a segment.

4. Bisect an angle.

5. Perpendicular to a line at a point on the line.

6. Perpendicular to line from point not on line.

7. Parallel to a line from a point not on line.

http://www.mathopenref.com/tocs/constructionstoc.html

Assignment: §5.1