An Introduction to Geometer’s Sketchpad In this lesson you will learn some of the basic features of Geometer’s Sketchpad. In particular you will learn how to draw points, circles, line segments, rays, and lines. The Measure menu will be also be explored, including how to measure angles, lengths, perimeter, and also use the Geometer Sketchpad’s built-in calculator. Finally, you will learn how to choose options from the Construct menu for objects such as parallel lines, perpendicular lines, midpoints, angle bisectors, etc. point: Locate and left-click on the point button which is located on the left side of your screen. The arrow that is controlled by your mouse should now have a red dot at the tip of it. Left-click anywhere in the drawing window and a point will appear. Draw several points in the window. How does the most recent point you have drawn look different from the previous points? circle: Locate and left-click on the circle button which is located on the left side of your screen. To draw a circle, left-click in the window. A point marking the center of the circle will appear. Now drag your mouse until the circle has the desired radius. Then left-click again. line: Locate and left-click on the line button which is located on the left side of your screen. If you click quickly, the default is to draw line segments. However, if you hold the left mouse button down while on the line button, a menu appears that allows you to choose between line segments, rays, and lines. Explore each of these options and draw a few line segments, rays, and lines in your window. select: Locate and left-click on the select button which is located on the left side of your screen (it is the arrow that points up and left). Left-click on some of the objects that you have already drawn (e.g. points, lines, and circles). What happens? How can you tell which objects are selected and which are not? If you select a line segment, does this action also select the endpoint vertices of the segment? What happens if you left-click an empty spot in the window while in select mode?

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The purpose of the select action is clear when we use the Measure and Construct menus. Measure menu: Locate the drop-down measure menu at the top of your window. You will notice numerous options. However, you may not be able to choose most of these. This depends upon which objects are, or are not, currently selected. Options such as Angle, Area, and Distance require that certain objects are selected before these options can be chosen. For example, to measure distance you can select two points. To measure an angle, select three points that define the angle. Experiment with some of the other options in the Measure menu and see which kinds of objects must be selected to use the option in question. What is the difference between distance and length? Caution: Often you may find that an option under the Measure menu is unavailable to you even though you selected the required objects. Usually this means that some extra things are selected that should not be. This is particularly problematic in complicated diagrams with a large number of objects. The best thing to do if this occurs is to left-click in an open area of your window. This un-selects everything. Then you can re-select only those objects that you need. Construct menu: Locate the drop-down construct menu at the top of your window. You will notice numerous options. As before, you may not be able to choose most of these, depending upon what is, or is not, currently selected. For example, draw a line segment in your window and make sure it is the only object selected. What options are available to you from the Construct menu? Try them and see what happens. i) To construct parallel and/or perpendicular lines, draw a line (ray or segment will also work) and a point not on the line (see picture below). Make sure both are selected. Go to the Construct menu and choose Parallel Line or Perpendicular Line. Are any other options available to you? If so, try them and see what they do.

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ii) Next draw two line segments that meet at a common vertex to form an angle. Make sure both line segments are selected. Go to the Construct menu and look for Angle Bisector. Notice that this option is not available to you. It appears that Geometer’s Sketchpad expects a different collection of objects to use this option. Figure out which objects need to be selected and then bisect your angle using the Angle Bisector. Go back to the Measure menu and measure your original angle. Then measure the two smaller angles to verify that your angle was indeed bisected. This activity could easily be done using a protractor and a compass. However, now let us see the advantage of dynamic geometry software. Left-click on one of the vertices from your original angle and start dragging it around the screen. In the upper left corner of your window you should see the angle measure changing as you move the vertex. In addition, the measure of the two smaller angles are also changing. What is always the relationship between these two angles and the original one?

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iii) Now draw a circle and plot two distinct points on the circle. Select both points and the circle. Choose Arc On Circle from the Construct menu. Notice that there are actually two different arcs that connect your points on the circle. Did you get the arc that you expected? How could you obtain the other arc? Now measure both the arc length and arc angle of your arc. What is the arc angle measuring?

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iv) Draw a circle in your window. With the circle selected choose Circle Interior from the Construct menu. Select and de-select the interior and notice the effect. With the interior selected measure the area inside the circle. Also, measure the radius and circumference of the circle. Change the radius of your circle by dragging the point on the circle (not the center) around the screen. Notice that the area and circumference are also changing as you do this. What would happen if you were to plot area vs. radius and circumference vs. radius from the data generated as you drag the point around? (Later we will see how to do these plots).

v) Explore the other options under the construct menu.

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Activity: Supplementary Angles. This exercise shows how Geometer’s Sketchpad can be used to make a simple conjecture. We also introduce the built-in Geometer’s Sketchpad calculator. 1) Draw a line segment in a new input window. 2) Draw a second line segment so that one endpoint of this segment lies on the interior of the first line segment:

3) Measure the angles m∠ABC and m∠DBC. Now we are going to use the calculator. Under the Measure menu choose Calculate... When the calculator is open left-click on m∠ABC which will be in the upper left corner of your window. In the calculator left-click on +. Then left-click on m∠DBC. Finally, choose OK on the calculator. At this point you should see the equation m∠ABC + m∠DBC = 180.00 under the two angle measures in the upper left corner of your window.

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4) Now drag the vertex C around. You should notice that m∠ABC and m∠DBC are changing as you do this. What is happening to m∠ABC + m∠DBC? Make a conjecture based on your observations. 5) You could also investigate supplementary angles using a straight edge and protractor. Is this Geometer’s Sketchpad activity more or less convincing (or the same) as the straight edge/protractor approach? Explain your answer.

Teacher Written Activity: Angle Sum for Triangles

Write a Geometer’s Sketchpad activity similar to the one above that investigates the interior angle sum for triangles. The activity should incorporate both the dynamic capabilities of Geometer’s Sketchpad as well as the built-in calculator.

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