NOTES TO TEACHER:This activity describes a step-by-step introductory lesson for students using Geometers Sketchpad. The objective is to calculate and compare the lengths of the sides of a triangle based on constructing circles and radii. The students use inductive reasoning to find a pattern and establish a postulate. If the students are experienced using Sketchpad then refer to the summary page and use it as a worksheet.
( fill-in all blanks )
Name_______________________ Date _____________________
(PART A)--Experiment 1
1. Choose PREFERENCES from the DISPLAY menu. Select Autoshow Labels for Points ( an X must appear in only this box ). While in the DISPLAY menu choose LABEL OPTIONS and mark the Increasing Part of Label : A. On the same screen, an X must appear in the box for Autoshow Labels for New Objects. Click OK.
2. Use the CIRCLE tool to draw 2 different size circles which intersect in 2 points.
NOTE: If the points labeled on the circles are too close to the point of intersection then use the ARROW (SELECT) tool to move them along the circle to another location.
3. Choose the ARROW ( SELECT ) tool and hold down the shift key to highlight each circle. Select POINT AT INTERSECTION from the CONSTRUCT menu. Now De-SELECT which means use the ARROW tool and click anywhere on the screen to un-highlight.
NOTE: If the points A, C, and E have different labels then highlight the point and choose Re-LABEL from the DISPLAY menu. Points B, F, and D will not be used.
4. Use the ARROW ( SELECT ) tool to highlight points A, C, and E. ( remember to hold down the shift key ). Choose SEGMENT from the CONSTRUCT menu. The three segments : EA, AC, and CE should appear. Then De-Select.
5. What familiar figure is drawn ? ______________________
6. With all three segments highlighted, choose LENGTH from the MEASURE menu. De-SELECT.
( record results )
EA= ____________ AC= ______________ CE=_____________
7. Now, compare the sum of any 2 lengths. Use the ARROW ( SELECT) to highlight any 2 numerical lengths and choose CALCULATE from the MEASUREmenu. A calculator will appear on the screen. Click Values and drag down to highlight one segment. Click the calculators + sign and then click on the other segment to be added. Both segments and the + sign must appear at the top of the calculator before clicking OK.
How does EA + AC compare to CE ? ______________________
How does EA + CE compare to AC ? ______________________
How does AC + CE compare to EA ? ______________________
8. Click on Point C ( the center of the circle ) and hold down the mouse button to slowly drag Point C.
What happens to the radii of the circles ?__________________
Are there any changes in the previous conjectures made in section 7 about comparing the lengths of the radii and the sides of the triangle ? ___________________________________________________
9. Choose SELECT ALL from the EDIT menu. Then choose CLEAR.
Draw 2 circles that do not intersect.
NOTE: If the points have different labels the highlight the point and choose ReLABEL from the DISPLAY menu. Use the ARROW(SELECT) to move a point on the circle.
10. Use the ARROW ( SELECT) and highlight point B and A. Choose SEGMENT from the CONSTRUCT menu. With the line segment highlighted, choose COLOR from the DISPLAY menu and then choose LENGTH from the MEASURE menu. De-SELECT and repeat this for points A and C but select a different COLOR (repeat for C and D with another color).
Record each length. BA=_______ AC=___________ CD=__________
Use the ARROW (SELECT) to highlight both BA and CD. Use the
calculator and compare.
Circle the answer : BA + CD > , < , = AC
WHY?__________________________________________________ (To help answer this question, click and drag Point D with the ARROW(SELECT) tool on the circle keeping the circles as non-intersecting. De-SELECT and do the same with Point B ).
( PART C )-- Experiment 3
11. Choose SELECT ALL from the EDIT menu. Then choose CLEAR. From the DISPLAY menu choose COLOR as black. Select the POINT tool and place 2 points on the screen.
12. Use the ARROW (SELECT) tool to highlight both endpoints and choose SEGMENT from the CONSTRUCT menu. With the segment highlighted, choose POINT ON OBJECT from CONSTRUCT menu. De-SELECT.
Re-label points if necessary by highlighting the point and choosing Re-LABEL from the DISPLAY menu.
13. Use the ARROW (SELECT) and shift key to highlight first point A and then point C. Choose CIRCLE with CENTER+POINT from the CONSTRUCT menu.
Now De-SELECT and repeat process with first highlighting point B
and then point C.
14. Draw a picture and label points.
What is true about the circles?_________________________
15. Use the ARROW ( SELECT) and highlight point A and C. Choose SEGMENT from the CONSTRUCT menu. With the line segment highlighted, choose LENGTH from the MEASURE menu.
De-SELECT and repeat this for points C and B.
De-SELECT and repeat this for points A and B.
Record each length. AC=_______ CB=__________ AB=__________
Use the ARROW (SELECT) tool to highlight both AB and BE. Choose the calculator and compare.
Circle the answer : AC + CB > , < , = AB
WHY?______________________________________________
( To help answer this question, click and drag Point C with the ARROW(SELECT) tool).
16. SUMMARIZE : THE IDEAS AS POSTULATES ( on the paper draw the segments for the three conditions in the previous experiments).
1). When is a triangle formed ? _______________________________
______________________________________________________
2). What is true about the lengths of radii in comparison to the length of the line segment that joins the center of the circles?
______________________________________________________
______________________________________________________
3). Name the theorem that justifies your results. What does it state ?
______________________________________________________
Diane McKenna and Arlene Riegel Scotch Plains-Fanwood H.S.- Scotch Plains, N.J. 07076
