## Learning Math: Geometry

# Parallel Lines and Circles Part A: Introduction to Geometer’s Sketchpad (25 minutes)

## Session 4, Part A

### In this part:

- Drawing with Geometer’s Sketchpad
- Tutorial
- Constructing with Geometer’s Sketchpad

Dynamic geometry software like Geometer’s Sketchpad allows you to construct and manipulate geometric figures. Such software can help you discover relationships among geometric figures, search for invariants (properties that do not change), and clarify the beginning stages of geometric proof. In this session, you will begin to use the tools of Geometer’s Sketchpad to construct geometric figures and explore their properties.

This session assumes you can do the following with Geometer’s Sketchpad: construct points and segments; construct points on objects; create midpoints; construct lines that are parallel or perpendicular to given lines; measure distances and angles; and perform calculations on your measurements. If not, or if you would like a refresher, please go to the tutorial.

As you learned in Session 1, there is a difference between drawing a figure and constructing a figure. If you are using a pencil and manual drawing tools, constructing requires that you use a compass and a straightedge. When you construct an object using dynamic geometry software, you use the software’s tools to give the figure the necessary characteristics.

**Problem A1**

Using Geometer’s Sketchpad, look at the following ways that you can construct a circle:

a. |
Use the Circle Tool to draw a circle. Notice that wherever you ended your circle, another point appears. The distance of this point from the center defines the radius of the circle. What happens when you move that point? What happens when you move the center point? |

b. |
Place two points anywhere on your sketch. Select them both, and construct a circle using the Construct menu. What happens if you select the points in the opposite order? |

c. |
Place a point anywhere on your sketch, and then draw a segment in a different location. Select both objects, and use the Construct menu to construct another circle. What does the point do? What does the segment do? Try moving both around and changing the size of the segment. |

Use the Circle Tool to construct a single circle. Then use the Circle Tool to construct a second circle, starting on your first circle and ending at its center. Make sure that you connect to the first circle’s center, or you may end up with two circles that only look like they go through each other’s centers.

**Problem A2**

Construct two circles that go through each other’s centers. See if you can construct them so that there are only two points that are displayed, as in the graphic below:

As you learned in Session 1, there is a difference between drawing a figure and constructing a figure. If you are using a pencil and manual drawing tools, constructing requires that you use a compass and a straightedge. When you construct an object using dynamic geometry software, you use the software’s tools to give the figure the necessary characteristics.

*The Geometer’s Sketchpad *® 2001, Key Curriculum Press, Emeryville CA, 1-800-995-MATH

### Tutorial

This section of Part A is designed to introduce you to The Geometer’s Sketchpad, a dynamic geometry software program from Key Curriculum Press. One of the major benefits of dynamic geometry Sketchpad and software like it is the ability to quickly alter geometric objects while preserving the properties of their construction. For example, you can use Sketchpad to create a triangle with one right angle, and then explore the relationship between the three side lengths by moving the triangle’s points around while preserving the constructed right angle. In this way, Sketchpad and software like it can allow users to discover geometric properties and come up with conjectures that will lead to proof.

Although this section is designed for new users of Sketchpad, users who are only familiar with an older version of Sketchpad may also find it useful.

**Downloading Sketchpad**

A limited version of Sketchpad 5 is available online at http://info.mheducation.com/sketchpad.trial.html. Please note that this software is copyrighted and subject to a Fair Use Agreement. Versions are available for Macintosh and Windows operating systems.

**Constructing with Tools**

When you start Sketchpad, you will see six tools on the left side of the window. These are as follows:

- Arrow Tool — Use this tool to select or drag objects.
- Point Tool — Use this tool to place points in the sketch.
- Compass Tool — Use this tool to create circles by center and radius.
- Straightedge Tool — Use this tool to create line segments.
- Text Tool — Use this tool to add captions and name points in a sketch.
- Custom Tool — Use to add more tools. (This tutorial will not use the Custom Tool.)

Two of the five major tools have a small arrow in their lower right corner. Clicking and holding the mouse button can change these tools:

Arrow Tool — can be used to select objects (default) or to rotate or translate objects

Straightedge Tool — can be used to draw line segments (default), rays, or lines

**Constructing a Triangle Using the Tools**

Here is a step-by-step process for how to use Sketchpad to construct a triangle:

- Select the Point Tool.
- Move the cursor into the sketch. You should see a red dot at the tip of the cursor, which represents a point you can place in the sketch.
- Click once to create a point. Move the mouse to another location, and again, click once to create another point. Repeat one more time. You should see three red dots in the sketch. The third point should have a purple highlight; this indicates that it is a selected object.
- Select the Straightedge Tool, and check that it is set to draw line segments (no arrows at either end).
- Move the cursor to the first point you placed in the sketch. You should see a blue highlight; this indicates that clicking the mouse button will start a line segment from this point.
- Click and hold down the mouse button from this point, and release the mouse button at the second point. You should see the second point briefly turn blue, and then a line segment will be created between the two points. It will appear with a purple highlight, which indicates that it is a selected object.
- Repeat steps 5 and 6 two more times to create the line segments, that complete the triangle.

**What If Something Goes Wrong?**

Chances are that if this is the first time you are creating a sketch, you will create points or line segments you did not intend to create. If this happens, you can delete any created object using this process:

- Select the Arrow Tool. This is very important! If the Arrow Tool is not selected, clicking on the sketch will create new objects.
- Click on the object you wish to remove. You should see a purple highlight, which indicates that the object has been selected.
- Hit the backspace key on the keyboard, or select “Clear” from the Edit menu.

If you start to construct something that you do not want, you can hit the backspace key in the middle of a construction to stop it.

**Construction Notes**

The Straightedge Tool can construct line segments, rays, or lines, depending on the active setting. Click and hold on the Tool to see the other settings. You can construct line segments, rays, and lines without first placing points; when this happens, the points will automatically be created in the sketch. When constructing rays, first select the endpoint of the ray, then a point along the ray.

It is often useful to construct a point on an existing object. For example, we might want to make sure a point stays on a particular line segment. You can create a point on an object in one of two ways:

- Select the Point Tool. Then move the cursor onto the object on which you wish to place a point. When the object shows a blue highlight, it means that the point will be placed on the object.
- Select the Arrow Tool. Click on the object, which will show a purple highlight. Then use the Construct menu to select “Point On….”

**Constructing a Point of Intersection**

First, clear your sketch. You can do this by using the Arrow Tool and deleting all objects. You can select all objects by dragging a bounding box around the entire sketch or by using “Select All” from the Edit menu. Then hit the delete key to delete the objects. You can also start a new sketch by selecting “New Sketch” from the File menu.

The following is a step-by-step guide to constructing the point at the intersection of two lines:

- Construct a line. Remember to change the Straightedge Tool first. Note that although the entire line appears immediately, the second point is placed only when you release the mouse button. It is a common mistake to assume that the second point has been placed already when creating a line that must pass through a certain point.
- Construct a second line that intersects the first line roughly in the center of the sketch.

Note that even though the lines intersect, there is no point (red dot) where the lines intersect. To construct this point there are two methods:

- Select the Point Tool. Move the cursor until both lines are highlighted in blue. You should see “… of Intersection” in the lower left corner of the window. Click the mouse to mark the point.
- Select both lines. To do this, select the Arrow Tool, click on one line, and then hold down the shift key and click on the other line. Both lines should now be highlighted in purple. Then, under the Construct menu, select “Intersection.” The point at the intersection of the two lines will then be created.

Be sure you are comfortable using both methods. The method of highlighting multiple objects using the shift key is useful and necessary in Sketchpad.

**Dragging Objects**

With the sketch you have created, select the Arrow Tool. Then click and hold down the mouse button on one of the original four points you created (in other words, any point other than the intersection point). Now, while holding down the mouse button, move the cursor around. You will drag the point and any other objects that rely on this point’s position. Here, moving a point will move the line through that point, and will move the point of intersection.

You can use this method to drag any object in a sketch. In this sketch, you can drag either line, and the line will remain parallel to its original position. You can drag the point of intersection, but since this point was constructed based on the rest of the sketch, doing this will move the entire sketch at once. Try dragging the objects around in this diagram. Is it possible to drag the objects so that the point of intersection disappears?

**Constructions Using the Menu**

Sketchpad provides a Construct menu for common constructions. We will now focus on constructing a perpendicular bisector to a line segment. Here is a step-by-step guide to do this, starting with an empty sketch:

- Select the Point Tool.
- Use the Point Tool to create two points in the sketch.
- Select the Arrow Tool.
- Select both points. Click once on one of the points so that it is highlighted (and it is the only object highlighted in the sketch). Hold down the shift key, and click once on the other point so that it is highlighted as well. Check that both points are in fact highlighted to indicate they are selected. If so, release the shift key, but do not click again in the sketch.
- From the Construct menu, select “Segment.” The line segment should be drawn and highlighted.
- From the Construct menu, select “Midpoint.” A point at the midpoint of the line segment should be drawn and highlighted.
- If the midpoint is already highlighted (it should be after construction), hold down the shift key and click once on the segment. Both the segment and midpoint should now be highlighted, indicating they are selected. If the midpoint is not already highlighted (if you clicked somewhere in the sketch), then click once on the midpoint to select it. Now follow the directions above.
- From the Construct menu, select “Perpendicular Line.” This line is the perpendicular bisector of the original line segment.

Note that the original two points can be dragged, and the perpendicular bisector will be moved corresponding to the new location of each point.

**Hiding Objects**

It is often useful to hide some of the objects that have been used to create points and segments. For example, try the following, beginning with the sketch of the perpendicular bisector:

- Create a point on the perpendicular bisector. To do this, select the Point Tool and move the cursor onto the line, which will highlight in blue.
- Select the Arrow Tool, and then click on the perpendicular bisector. It will highlight in purple.

At this point, if you hit the delete key to remove this object, the point you created on the perpendicular bisector would also disappear, since it was created based on the location of the perpendicular bisector. As an alternative, you can hide the line by selecting “Hide…” from the Display menu. The line will slowly disappear….

Important: When selecting objects, be careful! Actions such as “Hide” will affect all highlighted objects. It is very common to forget that an object is automatically highlighted when it is first created. To get around this, many Sketchpad users select the Arrow Tool, then click at an empty location in the sketch to make sure nothing has been highlighted before selecting a group of objects. This is especially important for certain types of constructions, which require points or lines to be selected in a specific order.

**Measurements and Calculations**

One other major feature of Sketchpad is the ability to measure segments, angles, and other geometric properties such as area. These tools can allow you to observe that a particular geometric property is invariant, or can allow you to develop a formula for the relationship between two quantities in a geometric figure.

Start by creating a triangle or completing the triangle using the three points in the previous sketch. To measure the length of a line segment, highlight the segment, then select “Length” from the Measure menu.

To measure an angle, highlight the three points of the angle in order. The second point highlighted must be the vertex of the angle. Remember that to select multiple objects, you must hold down the Shift key while selecting the objects.

Measures will be placed in the upper left-hand corner of the sketch and can be moved anywhere by dragging. (Remember to use the arrow tool to drag objects. Click on the object and hold down the mouse button while moving the mouse around.)

Additionally, selecting “Calculate…” from the Measure menu allows you to perform operations on existing measures. For example, you could use this to calculate the sum of the three angles in a triangle. First, measure the three angles, then go to “Calculate…” under the Measure menu. Once inside the calculator, you can click on the displayed angle measures (in your sketch). The measures will show up in the calculator window. So click the first measurement, then click “+”, then click the second measurement and “+” again. Click the third measurement and then “OK” to display their sum.

**Conclusion**

Continue to experiment with Sketchpad as you do the problems in this session. You should be able to do the following:

- Construct basic objects such as points, circles, and lines
- Construct points “on” objects and at the intersection of objects
- Select multiple objects using the shift key
- Drag objects and observe the updated sketch
- Make basic constructions using the Construct menu
- Hide objects using the Display menu
- Measure objects using the Measure menu and the “Calculate…” dialog box

### Constructing with Geometer's Sketchpad

A right triangle has one right angle. To construct a right triangle, you must construct two of its sides to be perpendicular. (Perpendicular segments intersect at a 90° angle.) If you construct a triangle to be right, then no matter how hard you try to stretch and change it by pulling on one of its vertices, it will always stay right.

To construct a perpendicular line, tell the computer what you want it to be perpendicular to and where you want it drawn. You need to select a segment (or a ray or a line) and a point before “Perpendicular Line” will be an option for construction. The software only constructs perpendicular lines, but you can create a segment on top of that line, then hide (don’t delete!) the line. (Select the line, then choose “Hide Line” under the Display menu.)

**Problem A3**

a. |
Draw a right triangle made of three line segments. “Eyeball” two of the line segments so that it looks like they meet at a 90° angle. |

b. |
Construct a right triangle. Notice that it would be difficult to tell which one was “drawn” and which one was “constructed” just by looking at your triangles. |

c. |
Grab each of the vertices of the first triangle, one at a time, and drag them around. What changes? What stays the same? |

d. |
Grab each of the vertices of the second triangle, one at a time, and drag them around. What changes? What stays the same? |

### Solutions

### Problem A1

a. |
Moving the point on the circle changes the radius. Moving the center changes the radius and rotates the circle about a point on the circle. |

b. |
The point chosen first is the center of the circle. |

c. |
The point determines the center and the segment determines the radius. |

### Problem A2

Draw a circle. Two points will be displayed: the center and a point on the circle. Now construct another circle using the point displayed on the first circle as the new circle’s center. Move the mouse to the center of the first circle and click.

### Problem A3

a. |
Follow the instructions. |

b. |
Start by creating a segment anywhere on your sketch. Then select the segment and one of its endpoints. (Remember to hold down the shift key while selecting more than one object.) When you have a segment and a point selected, “Perpendicular Line” will be an option under the Construct menu. Create a point anywhere on the perpendicular line. Then select the perpendicular line and choose “Hide Line” under the Display menu. Connect the point to each endpoint of your original segment. |

c. |
Everything can change depending on what you select — sides, angles, etc. Nothing remains fixed. |

d. |
In this case, the right angle remains fixed no matter what you do. |