## Learning Math: Geometry

# Parallel Lines and Circles Part D: Thinking with Technology (15 minutes)

## Session 4, Part D

Now that you’ve had some experience with dynamic geometry software, discuss or reflect on the following problems.

### Problem D1

No geometry software will allow you to construct parallel lines or perpendicular lines without beginning with both a line and one point. Why?

### Problem D2

What are the advantages of constructing a figure in dynamic geometry software instead of just drawing something that looks like it has the characteristics you require?

### Problem D3:

How did using dynamic geometry software help or hinder your learning of the topics presented in this session?

### Solutions

**Problem D1**

In either construction, the software is expected to draw a particular line. There are, however, an unlimited number of lines parallel or perpendicular to a given line. (A sheet of graph paper provides a good example of this.) To draw a particular line, a point must also be provided to pick one line out of the many possible parallel or perpendicular lines.

**Problem D2**

Answers will vary. One of the advantages is the precision with which we can construct figures. Another one is that the measurement tools combined with the ability to modify figures dynamically allow us to observe changes and make conjectures about what aspects of the figures are invariant. Most importantly, you can test several hundred cases with just one construction. For example, if you want to know something about right triangles, you could draw numerous cases, or you could construct just one and change it into all those cases you drew (and more). When we construct something, we know the results are true for all cases. Nevertheless, these would still be conjectures, and conjectures must be proved.

**Problem D3**

Answers will vary.