Guidance
As you have studied math, you have often created drawings . Drawings are a great way to help communicate a visual idea. A construction is similar to a drawing in that it produces a visual outcome. However, while drawings are often just rough sketches that help to convey an idea, constructions are step-by-step processes used to create accurate geometric figures.
Constructions take us back over 2000 years to the ancient Greeks, before computers or other advanced technology. Using only the tools of a compass and a straightedge , they discovered how to copy segments, angles and shapes, how to create perfect regular polygons, and how to create perfect parallel and perpendicular lines. Today, learning constructions is a way to apply your knowledge of geometric principles. You can do constructions by hand, or with dynamic geometry software. In this concept, the focus is on hand constructions and making copies of segments, angles, and triangles.
To create a construction by hand, there are a few tools that you can use:
- Compass: A device that allows you to create a circle with a given radius. Not only can compasses help you to create circles, but also they can help you to copy distances.
- Straightedge: Anything that allows you to produce a straight line. A straightedge should not be able to measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only use it to draw straight lines and not to measure.
- Paper: When a geometric figure is on a piece of paper, the paper itself can be folded in order to construct new lines.
Example A
Use a straightedge to draw a line segment on your paper like the one shown below. Then, use your straightedge and compass to copy the line segment exactly.
Solution: First use your straightedge and pencil to create a new ray.
Now, you have one endpoint of your line segment. Your job is to figure out where the other endpoint should go on the ray. Use your compass to measure the width of the original line segment.
Now, move the compass so that the tip is on the endpoint of the ray.
You can now see where the endpoint of the segment should lie on the ray. Draw a little arc with the compass to mark where the endpoint should go.
You can use your straightedge to draw the copied line segment in a different color if you wish.
Note that in this construction, the compass was used to copy a distance. This is one of the primary uses of a compass in constructions.
Example B
Use a straightedge to draw an angle on your paper like the one shown below. Then, use your straightedge and compass to copy the angle exactly.
Solution: Keep in mind that what defines the angle is the opening between the two rays. The lengths of the rays are not relevant.
Start by using your straightedge and pencil to draw a new ray. This will be the bottom of the two rays used to create the angle.
Next, use your compass to make an arc through the original angle. It does not matter how wide you open your compass for this.
Next, leave your compass open to the same width, and make a similar arc through the new ray.
Now, you know that the second ray necessary to create the new angle will go somewhere through that arc. Measure the width of the arc on the original angle using the compass.
Leave the compass open to the same width, and move it to the new angle.
Make a mark to show where the pencil on the compass intersects the arc.
Use a straightedge to draw another ray that passes through the point of intersection of the two compass markings.
You have now copied the angle exactly.
Example C
An angle is created from two line segments. Use a straightedge to draw a similar figure on your paper. Then, use the straightedge and compass to copy the figure exactly.
Solution: To copy this figure, you will need to copy both the line segments and the angle.
Start by copying the line segment on the bottom using the process outlined in Example A (draw a ray, use the compass to measure the width of the line segment, mark off the endpoint on the ray).
Next, copy the angle using the process outlined in Example B (draw an arc through the angle and draw the same arc through the new ray, measure the width of the arc, draw a new ray through the intersection of the two markings).
Finally, copy the second line segment by measuring its length using the compass and marking off the correct spot for the endpoint.
You can now draw the copied segments in a different color for emphasis.
Vocabulary
A drawing is a rough sketch used to convey an idea.
A construction is a step-by-step process used to create an accurate geometric figure.
A compass is a device that allows you to create a circle with a given radius. Compasses can also help you to copy distances.
A straightedge is anything that allows you to produce a straight line. A straightedge should not be able to measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only use it to draw straight lines and not to measure.
Guided Practice
You drew a triangle similar to the one below for the concept problem.
- Copy your triangle using by using line segments.
- Copy your triangle using two line segments and an angle.
- Copy your triangle using two angles and a line segment.
Answers:
1. Start by copying one line segment. Here, the base line segment is copied.
Next, use the compass to measure the length of one of the other sides of the triangle.
Move the compass to the location of the new triangle and make an arc to mark the length of the second side of the triangle from the correct endpoint.
Repeat with the third side of the triangle.
The point where the arcs intersect is the third vertex of the triangle. Connect to form the triangle.
Note that with this method, you have only used the lengths of the sides of the triangle (as opposed to any angles) to construct the new triangle.
2. Start by copying one line segment. Here, the base of the triangle is copied.
Next, copy the angle at one of the endpoints of the line segment.
Copy the second side of the triangle (that creates the angle you copied) onto the ray that you just drew.
Connect to form the triangle.
3. Start by copying one line segment.
Next, copy the angle at one of the endpoints.
Copy the angle at the other endpoint of the line segment.
Connect to form the triangle.
Practice
- What is the difference between a drawing and a construction?
- What is the difference between a straightedge and a ruler?
- Describe the steps for copying a line segment.
- Describe the steps for copying an angle.
- When copying an angle, do the lengths of the lines matter? Explain.
- Explain the connections between copying a triangle and the triangle congruence criteria.
- Draw a line segment and copy it with a compass and straightedge.
- Draw another line segment and copy it with a compass and straightedge.
- Draw an angle and copy it with a compass and straightedge.
- Draw another angle and copy it with a compass and straightedge.