Watch this video lesson to learn about geometric construction and how you can copy line segments and angles without using any numbers. All you need is a straight edge and a compass.
What is geometric construction? It is the drawing of lines, angles, and shapes using only a pen or pencil, compass, and a straight edge. There are no numbers you have to deal with.
Why is this useful? It is useful when you have to draw lines and angles without measuring anything. And if you are an artist, this is a handy skill to have to ensure that any lines or angles that you copy are exactly the same.
The only tools you will need to copy and construct any line segment or angle, along with your pen or pencil, are a compass and a straight edge. In today's world, the most common straight edge is the ruler. But you can use anything that will give you a nice clean straight line when you take your pencil and run it along an edge. The compass is the mathematical tool that lets you draw nice clean circles and arcs; it's not the compass that people use to navigate directions.
Copying a Line Segment
Let's see what it takes to copy a line segment with just these tools. If you have a compass and a straight edge, follow along on your own piece of paper.
So, to copy a line segment, we begin with a line segment, which we will call line segment AB. Then, we will draw a point that is not on the line segment. We will call this point C. Now, we are going to put one end of the compass on point A and extend the compass so the other end is at point B. The compass is now the exact length of line segment AB.
We now move the compass so that one end is on point C. With the other end, we draw a small arc so we know where the line segment should end. We then mark a point on the arc to act as our end point. Then, we take our straight edge and connect point C to the end point. We have just copied a line segment using just a pencil, compass, and straight edge.
Exact line segment
Copying an Angle
Now, what about copying an angle? Follow along if you can with this one, too.
We start with our angle BAC. We draw a point D somewhere away from the angle. Then we take our straight edge and draw a line from point D. This line can be any length, and it can go in any direction, as long as it starts from point D.
Now we take our compass and place one point on point A. The width can be set to anything. We now draw an arc intersecting both lines of our angle BAC. Keeping the width of the compass the same, we move the compass to point D and draw another arc of similar size, making sure it intersects the line coming from point D on one end.
Now, we move our compass back to our original angle and place one end on one of the intersections formed by the arc. We change the width of the compass so the other end is at the other intersection. With this new width, we move the compass to the angle we are about to draw and place one end where the arc intersects the line. With the other end, we draw a small arc that intersects the previously drawn arc.
Creating an exact angle
This new intersection of the small arc with the bigger arc we will call point E. We finish our angle by taking our straight edge and connecting points D and E. We can extend our line beyond point E if we wanted to.
What have we learned? We learned that geometric construction is about drawing lines, angles, and shapes with only a pencil, compass, and a straight edge. It uses no numbers and no measuring. When it comes to copying a line segment or angle, all that is needed is to make arcs and intersections to mark the proper distances and locations of end points and such.
After completing this lesson, you should be able to:
- Construct a matching line segment using only a straight edge and compass
- Match an exact angle using the same straight edge and compass