8.1 Technology: Constructing Loci Using The Geometer s Sketchpad

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1 8.1 Technology: Constructing Loci Using The Geometer s Sketchpad A locus is a set of points defined by a given rule or condition. Locus comes from Latin and means place or location. An example of a locus is the set of points that are all the same distance from a fixed point. This locus is a circle, where each point on the circumference is the same distance from the fixed point. The fixed point is the centre, and the distance is the radius. In this section, The Geometer s Sketchpad will be used to construct several loci. Points Equidistant From Two Given Points You are given a line segment AB. Follow the steps below to find the locus of points that are equidistant from points A and B. Using The Geometer s Sketchpad, circles can be used to find the points for this locus. 1. Construct a line segment AB in the lower part of the screen. 2. Construct a line segment CD near the top of the screen. 3. With line segment CD selected, choose Point On Object from the Construct menu. Label this point E. 4. Select points C and E. From the 5. Select point A and line segment CE. By Center and Radius. 6. Select point B and line segment CE. By Center and Radius. 588 MHR Chapter 8

2 7. Select both circles. From the Construct menu, choose Point At Intersection. Label these points F and G. 8. Select points F and A. From the Measure the length of FA. Construct FB and measure its length. Are these lengths equal? If so, then point F satisfies the definition of the locus. Are the lengths of GA and GB also equal to these lengths? If so, then point G also satisfies the condition of the locus. 9. Select points F and G. From the Display menu, choose Trace Points. 10. Select the two circles. From the Display menu, choose Hide Circles. 11. Drag point E back and forth along line segment CD. The path of the points F and G represents the locus of points that are equidistant from point A and point B. 12. Select point F and point G. From the 13. Select line segments FG and AB. From the Construct menu, choose Point At Intersection. Label this point H. 14. Measure AHF, AHG, BHF, and BHG. Drag point A, point B, or point E to a new location. Do the measures of these angles change? 15. Measure line segments AH and BH. How do the lengths compare? Will they always compare in this way? 16. Describe the locus of points obtained from this construction. To test the construction and your description, drag point A or point B to a new location and create the new locus. 8.1 Technology: Constructing Loci Using The Geometer s Sketchpad MHR 589

3 17. Extension a) Select Point E and line segment CD. From the Edit menu, choose Action button and sub-option Animation. Change the first and third drop boxes, as shown in the window. Click on the Animate button. A new button,, will be created on the screen. Double click this button. Point C will be dragged along the line segment and the locus will be left on the screen. To test the construction, drag either of points A or B to a new location and double click on the Animate button. b) Select points F and G. From the Display menu choose Trace Points. This will remove the checkmark beside Trace Points, so that this feature is deactivated. Select point F and point E in that order. From the Construct menu, choose Locus. This will cause point E to be moved along line segment CD. The various locations of point F will not only be displayed on the screen, but will remain there. Repeat this process but select point G and point E. Drag one of the endpoints of AB to a new location, and describe what happens. Points Equidistant From the Sides of an Angle You are given BAC, formed by drawing two rays AB and AC. Point D is on AB and point E is on AC, so that AD = AE. Use the following steps to find the locus of points that are equidistant from points D and E. 18. Click the Line Tool on the toolbox and hold the left mouse button down. A window will open, giving a choice of a line segment, a ray, or a line. Choose the ray, and construct ray AB in the same way as a line segment is constructed. Construct a second ray AC. 19. Select ray AB. From the Construct menu, choose Point On Object. Label this point D. 20. Select points A and D, in that order. by Center and Radius. 590 MHR Chapter 8

4 21. Select the circle and ray AC. From the Construct menu, choose Point At Intersection. Label the new point E. Now, points D and E are the same distance from the vertex, A. 22. Select the circle. Choose Hide Circle from the Display menu. 23. Create a line segment FG at the bottom of the screen. With line segment FG selected, choose Point on Object from the Construct menu. Label this new point H. 24. Select points F and H. From the 25. Select point D and line segment FH. By Center And Radius. 26. Select Point E and line segment FH. By Center And Radius. 27. Select the two circles. From the Construct menu, choose Point At Intersection. Label these points I and J. 28. With points I and J selected, choose Trace Points from the Display menu. 29. Drag point H along line segment FG. The locus of points forms a line passing through points A, I, and J. 30. Select points J, A, and D in that order. From the Measure menu, choose Angle. Select points J, A, and E in that order. From the Measure menu, choose Angle. How do the measures of the angles compare? Will they always compare in this way? 31. Describe the locus of points created by this construction. To test the construction and your description, drag any of points A, B, or D to a new location, and drag point H along FG. 8.1 Technology: Constructing Loci Using The Geometer s Sketchpad MHR 591

5 32. Extension a) Select point H and line segment FG. From the Edit menu, choose Action Button. Choose the Animation sub-option. Change the first drop box to once and the third drop box to slowly. Click on Animate. A new button,, appears on the screen. Double click this button to get the locus of points. b) Click points J and I. From the Display menu, choose Trace Points. This will deactivate the trace feature for these two points. Select point J and point H in that order. From the Construct menu, choose Locus. Select point I and point H. From the Construct menu, choose Locus. The locus of points traced out by points I and J, as point H is dragged along FG, is shown and remains on the screen. Increase and decrease the measure of the original angle by dragging point C or point B. Does the construction work for an obtuse angle? Points Equidistant From Two Parallel Lines To construct the locus of points that are equidistant from two parallel lines, we will first construct a point on each line so that the line segment joining the points is perpendicular to the parallel lines. 33. Construct a line through points A and B. Make sure the Line Tool is selected in the toolbox at the left side of the screen. Construct a point C above the line. 34. Select point C and the line through A and B. From the Construct menu, choose Parallel Line. 35. With the new line selected, choose Point On Object from the Construct menu. Label this point D. 36. Select point D and the line through A and B. From the Construct menu, choose Perpendicular Line. 37. Select this new line and the line through A and B. From the Construct menu, choose Point At Intersection. Label this point E. 38. Select the line through D and E. From the Display menu, choose Hide Line. 39. Select points D and E. Change the Line Tool to line segment. From the 592 MHR Chapter 8

6 40. Construct a line segment FG at the bottom of the screen. With line segment FG selected, choose Point On Object from the Construct menu. Label this point H. 41. Select points F and H. From the 42. Select point D and line segment FH. From the Construct menu, choose Circle By Center and Radius. 43. Select point E and line segment FH. From the Construct menu, choose Circle By Center and Radius. 44. Select both circles. From the Construct menu, choose Point At Intersection. Label the points I and J. With the two points selected, choose Trace Points from the Display menu. 45. Drag point H along line segment FG. 46. Describe the locus of points. To test the construction and your description, drag any of points A, B, or C to a new location, and drag point H again. 47. Extension a) Select point H and line segment FG. From the Edit menu, choose Action Button. Choose the Animation sub-option. Change the first drop box to once and the third drop box to slowly. Click on Animate. A new button,, appears on the screen. Double click this button to get the locus of points. b) Click points J and I. From the Display menu, choose Trace Points to remove the trace feature. Select point J and point H in that order. From the Construct menu, choose Locus. Select point I and point H. From the Construct menu, choose Locus. The locus of points traced out by points I and J, as point H is dragged along FG, is shown and remains on the screen. To test the construction, drag A or B to a new location to change the slope of the original parallel lines. Dragging point C changes the original distance between the parallel lines. 8.1 Technology: Constructing Loci Using The Geometer s Sketchpad MHR 593

8.3 Technology: Loci and Conics

8.3 Technology: Loci and Conics The diagram shows a double cone. The two cones have one point in common. The intersection of a double cone and a plane is called a conic section or a conic. The circle,