Using Cabri Geometry Final Part A Problem 1

Transcription

1 Using Cabri Geometry Final Part A Problem 1 By: Douglas A. Ruby Date: 11/10/2002 Class: Geometry Grades: 11/12 Problem 1: Use Cabri-Jr. to create a triangle with vertices labeled A, B, and C. Then: a) Show that an exterior angle is equal to the sum of the opposite interior angles. b) Construct a parallel line to one of the sides of the triangle and show that the alternate interior angles and corresponding angles are equal. Use TI Connect to import your graphs into a word document and document your procedures. 1. Basics of using the TI-83+. Ok, we are going to start this class by using Cabri Jr. on the TI-83+ to construct and measure geometric figures. To start with, let s look at the basic TI-83+ keyboard. Your keyboard should look like this: Arrow Keys Final Part A Problem 1 Doug Ruby - Page 1

2 Start by hitting the ON key at the lower left of the keypad. Next, let s set ourselves up to run Cabri Jr. To do this, let s hit the APPS key next to the MATH key. You should see the following window: Use the Down Arrow (the purple one in the upper right quadrant) to position the blinking cursor to the 2: in front of Cabri Jr and then hit Enter. You will then see: Now hit any key to actually enter the Cabri Jr. program. While you initially may get a blank screen, you will shortly be presented with a screen that says: This is telling you that there are menu functions attached to the F1, F2, F3, F4, and F5 keys. Before we start, however, let s define the problem we want to solve. Page 2

3 2. Analyzing the interior angles of a triangle We will use Cabri Jr. to create a triangle labeled ABC. We will then measure the interior angles of ABC. Further, we will create an exterior angle, measure that angle, and compare it with the sum of the remote interior angles. Finally, we will create a line parallel to one side of the triangle through the opposite vertex. For example, in ABC, we would could create a line parallel to segment AB through point C to achieve our ends. We then will show that the alternate interior angles and corresponding angles are of this parallel line are equal. 3. Create a triangle Initially, we want to use the F2 function to construct geometric elements such as points, lines, segments, circles, triangles, or quadrilaterals. As you can see, there are a number of options. Using the arrow keys, scroll down to Triangle. We then hit Enter. Once you have selected the triangle tool, you see little triangle in the upper left corner of the screen. Now you should see the following: will The pencil in the screen represents the drawing tool that will create our triangle. To create a triangle, move the pencil to the point where you want the first vertex of your triangle, then hit ENTER. This will create one of the three vertices of your triangle. Then use the arrow keys to move the pencil towards the location of the second vertex of your triangle. As you move the pencil towards the second vertex, you should see a dotted line begin to extend from the first point as in the picture to the left. Continue to use the arrow keys until the pencil is at the point you want for your second vertex, then hit ENTER. Page 3

4 Now that we have located two the three vertices of your triangle, we need to create the third vertex, thus completing the triangle. To do this, continue to use the arrow keys to move the pencil towards the location you want for your third vertex. Your screen should look like that on the left. Once you are where you want the third vertex to be, hit ENTER. This will create the third vertex. Your triangle should now be complete and have solid instead of dotted lines. 4. Label your triangle. Now we want to label the triangle we just created. To do this, we hit the F5 button on the calculator. This brings up a menu. We use the arrow keys and ENTER to select the Alph-Num menu item. Once w have done this, we should see the text bar cursor. This looks like the following: Notice that the bar cursor is positioned over the last point created by the triangle tool. If we hit ENTER now, we can start writing where the bar cursor is. Hit the ALPHA (green) key on your TI-83+ and then enter A. (note that the A is the green character on the MATH button. The label now appears. Hit ENTER again. The screen should now look at it does on the left. If you want to move the A so it can be seen more clearly, we need to get into the mode where we can move objects on the screen. To do this, hit the TI-83+ CLEAR button. You will now see an arrow. Position the arrow directly on the A (being careful not to be on the point under the A.) The arrow should be see through instead of black. Hit ALPHA. You should now see a hand. Using the arrow keys, move the character A to the left until it is not obscured by the lines and vertices of the triangle then hit ENTER. You now should see the following: Page 4

5 Continue to use the F5-Alph-Num text tool and object movement tools until your triangle is clearly labeled as in the screen shot to the left. 5. Create exterior angles Now that we have created and labeled triangle ABC, we need to create the exterior angles. To do this, we must create lines that extend through the vertices of the triangle. We use the F2-Line tool. Once we have selected the line tool, it will drop the pencil tool onto our screen. We move the pencil until it is over one of the vertices of the triangle. We then hit ENTER. We then move the pencil (and this the 2 nd point that constructs the line) until it is over another vertex of the triangle. Hit ENTER again. This will have constructed a line (as opposed to a segment) that goes through two vertices of triangle ABC. We continue to repeat this process until we have created three lines that intersect the vertices of triangle ABC as in the screen shot directly above. This process of extending te sides of the triangle has the effect of creating all of the exterior angles of our triangle. 6. Measure exterior angle Now that we have constructed our exterior angles, we need to measure one of them. In this case, lets measure the angle exterior to vertex A. To measure an angle, we must select three points, much as in Geometer s Sketchpad. However, we have only two points (A and C) with which to measure the angle exterior to A. Thus, we need to create a point on line AB that is exterior to triangle ABC. To do this, use the F2-Point>Point-on tool. This will allow us to create a Point on line AB. Once the arrow is near line AB, it will change to a pencil. This means that you can hit enter to locate the point on line AB. (as in the screen to the left). Knowing that this point is on the line, we now use the F5-Measure>Angle tool to measure the exterior angle. First select the exterior point and hit ENTER. Then move the vertex of the angle (point A), and hit enter. Finally move to another point (Point B) and hit ENTER. With three points selected, the measurement of the angle should appear near the vertex. In this case, it is 131 o. Page 5

6 7. Measure opposite interior angles Now that we have measured the exterior angle (in this case 131 o ), we need to measure the two opposite interior angles whose vertices are point B and C of the triangle (respectively). We use F5- Measure>Angle again to do this. In our example, these angles are 85.6 o and 45.4 o. The sum of these two angles is also 131 o and thus we have demonstrated that the exterior angle is equal to the sum of the opposite interior angles. 8. Demonstrate an oddity in Cabri-Jr. As we have experimented with Cabri-Jr, the fact that it limits measurements to 3 significant digits provides us with the opportunity to demonstrate an oddity. Having measured the interior angles at vertices B and C, we use F5-Measure>Angle again to measure vertex angle A. We find this to be 48.9 o. Unfortunately, if we add the three interior angles of ABC as measured by Cabri-Jr, we find that they add up to o. This is an unfortunate byproduct of the fact that Cabri-Jr is limited in its ability to display significant digits and appears to round digits. However, the internal representation of these numbers must be more accurate than the results displayed, since we can use F5-Calculate to let Cabri-Jr calculate the Triangle Sum using its internal representation of the numbers displayed. In the diagram above right, we have done this and find that this sum is 180 o exactly as displayed in the lower right hand corner of the screen. To further demonstrate this characteristic, We calculated the sum of the opposite interior angles using the F5-Calculate function and found this sum to be o as opposed to 131 o. However, in a further test, if we subtracted the measure of the exterior angle, the final result as 0 o. This seems to prove that rounding and lack of precision in the numeric display (all results of the small screen pixel resolution) are behavioral characteristics that may confuse the user of Cabri-Jr. Page 6

7 9. Create a parallel line through a vertex In order to create a line parallel to line AC through vertex B, we need to use the F3-Parallel tool. As one cane see to the left, this pop-up appears and we use the scroll keys to select the parallel line tool. Notice the two parallel lines appear in the upper left box on the screen to tell us we are using the parallel line tool. Once we hit ENTER a selection cursor appear on the screen. When we move this cursor over line AC, it changes to a solid block arrow as in the screen below: Once the cursor is positioned over the line we want the new line to be parallel to, we hit ENTER. The cursor now changes to a pencil and we can see a dotted line that can be moved around to any position we want. As we start to use the arrow keys to move the line upwards towards vertex B, we will see something like the screen to the left. In order to insure that this line passes exactly through point B, we MUST position the pencil directly over point B. In the screen to the right, we have positioned the pencil directly over point B. If we look carefully, we will see that the little circle that represents point B is actually flashing. Once the pencil is over the point we want to drop the parallel line through, we hit ENTER. We now have created a line that passes directly through point B that is parallel to line AC. Page 7

8 10. Place points on the parallel line In order to measure the alternate interior and corresponding angles created by the new line parallel to AC that passes through B, we must drop points on this line. To do this, we use the F2- Point>Point_on tool as suggested to the right. Once we have selected this tool by hitting ENTER, the block arrow or pencil cursor should appear. If the cursor is in free space (i.e. not near a selected object), then you will see a block arrow cursor. If the cursor is a pencil, then it is near enough to an object for that object to be the one the point will be constructed on top of. In this case we position the cursor to the left of point B on the new parallel line as suggested in the screen to the right. Then we hit ENTER. A point will be created on the new parallel line. Next we move the cursor to the right of point B and create another point on the parallel line. This second point is pictured to the left. If we move the cursor away from any object, we can see that it changes back to the block arrow instead of the pencil. We also need to create exterior points on the extended lines AB, BC. This is pictured to the right. 11. Measure alternate interior and corresponding angles. Now that we have created reference points on lines so that we can make angle measurements, we select the F5-Measure>Angle tool as indicated to the left. Page 8

9 Looking first at the alternate interior angles, we measure the angles that alternate with the interior vertex angles A and C (48.9 o and 45.4 o respectively). As you can see below, these alternate interior angles created by the parallel line through B are the same measures as the interior vertex angles. Next, we measure the angles that correspond to interior vertex angles A and C. In this picture, one can see that we have measured the angles that correspond to the vertex angles A and C. Note that they are off by 0.1 o from our prior measurements. Unfortunately, with Cabri-Jr, this may be as good as it gets. Notice that the two points on lines AB and CB near the top of the screen are near point B. I believe that the internal (X, Y) coordinates of those points are based on pixel resolution of TI- 83+ screen. Due to the relative closeness of these points to vertex B of our triangle, the accuracy of the measurement of these two angles is slightly off. Conclusion We have demonstrated the use of Cabri-Jr to create a triangle with vertices labeled A, B, and C. We showed that an exterior angle is equal to the sum of the opposite interior angles. We constructed a parallel line to one of the sides of the triangle and within the measurement error of Cabri-Jr demonstrated that the alternate interior and corresponding angles are equal to the opposite interior angles. We also demonstrated some of the limitations of Cabri-Jr. I believe that these limitations, driven mostly by the lack of screen resolution on the TI-83+, significantly hamper the usefulness of Cabri-Jr in a classroom environment. Page 9

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