## The Geometer's Sketchpad Introductory Lab - page 5

### Mike Riedy

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By this time you should be able to draw pretty pictures, but what about the real geometry involved?

#### Measurements:

Start with a new sketch. Draw a segment, a line, and a circle. Highlight the segment but not the endpoints. What about the segment can you measure, and what are the measurements (do not include "Calculate")?
```        What can be measured?	        What is the measurement?

________________________        ________________________

________________________        ________________________

```
Now highlight just the endpoints but not the actual segment. What about the segment can you measure, and what are the measurements (do not include "Calculate")?
```        What can be measured?	        What is the measurement?

________________________        ________________________

________________________        ________________________

```
Highlight just the circle. What about the circle can you measure, and what are the measurements (do not include "Calculate")?
```        What can be measured?	        What is the measurement?

________________________        ________________________

________________________        ________________________

________________________        ________________________

________________________        ________________________

```

#### Constructions:

There are two ways to construct things with GSP. The first is to draw objects free-hand by using the basic tools in the toolbox. The second method is to use the CONSTRUCT menu and all its possible choices. The second method is more accurate and should be used whenever possible.

Let's try a few constructions. We will begin by playing with the circle. Highlight the circle and choose "Point On Object" under the CONSTRUCT menu. A point should appear on the circle. Highlight the new point and the center of the circle, but not the circle itself. Tell the computer to construct the segment connecting those two points.

What is this special segment called? ____________________________

What is the length of this segment? ____________________________

Now highlight only the circle and construct the "Circle Interior." Change the color of the interior to green by looking under the DISPLAY menu for the choice of "Color".

Let's play with the segment for a while. Highlight the segment and construct a "Point At Midpoint." Highlight the midpoint you just constructed and the segment itself. Construct the "Perpendicular Line." Change the color of this new line to blue.

If you want to know how to use any of the other CONSTRUCT menu choices you can choose "Construction Help" under the CONSTRUCT menu. This will explain everything you need to know about constructions. Save this sketch to your disk as "Construction" (see p. 5 for a refresher on saving).

#### Angles:

One of the building blocks of geometry is the angle. We are now going to look at some things you can do with angles using GSP.

Start with a new sketch (don't forget the keyboard shortcut). An angle is defined as "two rays that share a common endpoint." Using this definition, construct an angle on your new sketch. Make sure you use rays and not segments or lines. Also make sure the two rays share their endpoints.

If your construction is correct, you should be able to see three distinct points: outside points (one on each ray), and the common endpoint, called the vertex. Drag any of the three points to make sure you have an angle.

Display the labels for the three points (see p. 4 for a refresher on labeling). Now construct a segment connecting the outside points by highlighting both points only   and choosing "Segment" under the CONSTRUCT menu (make sure the segment tool is showing in the toolbox). Highlight the three points in the following order:

1. an outside point
2. the vertex
3. and the other outside point.
Now measure the angle.
How big is the angle? ______________________

What does the little symbol after the angle measure stand for?

__________________________________________

Clear all the highlighting by clicking any blank area on your sketch. This is another trick commonly used and very important when using GSP.

Highlight the three points again but in a different order with one of the outside points as the second chosen point.

What is the measure of this angle? ______________________

Now repeat this procedure with the other outside point as the second chosen point.

What is the measure of this angle? ______________________

Now let's do a little calculating. We want to find the sum of the three angles from above. You could do the calculation in your head or even pull out your handy-dandy pocket calculator, but wait - there is an easier way! GSP will do the calculation for you.

Highlight the three angle measures in the upper left-hand corner of your sketch. Choose "Calculate" under the MEASURE menu. Under the bar that reads "Values" you will find the names of all three angles you measured. Choose one of the angles, "+", another angle, "+", and the last angle. Then click "OK".

What is the sum of the three angles? ______________________

You should find the sum to be greater than 150 degrees but less than 200 degrees. If not, try again.

Drag one of the outside points.

What changes as you drag the point? ______________________

What does not change? ______________________

You just measured the three angles of a triangle. What can you conclude about the sum of the measures of the angles of a triangle?

_____________________________________________________________

_____________________________________________________________

Save the angle sketch to your disk as "Angle sums" (see p. 3 for a review of saving procedures).

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