Ellipses

1. Open the ellipse.gsp file in Geometer's Sketchpad. Click the animate button. What do you see?
2. What criteria did we use to fold the ellipse in the previous activity? Is this a definition? How can we prove the ellipse on the screen follows the same rule?
3. Create a segment from the point that traces the ellipse to the moving point on the circle.
4. Create another segment from the point inside the circle to the point that traces the ellipse.
5. Construct a third segment that makes a triangle. Select this segment and construct its midpoint.
6. Construct a segment from the point that traces the ellipse to the midpoint.
7. Using what you know about basic geometry, can you prove that you are really making an ellipse? Look back at the definition in number 2, and animate the drawing again if necessary.
8. Hide all of the segments you just made, and the midpoint.
9. We will now call the center of the circle and the point that is inside the circle the foci of the ellipse.
10. Construct two segements that represent the distance from the point that traces the ellipse to each of the foci.
11. Use the measure --> length command to display the lengths of each of these segments on the screen.
12. Move the point on the circle around the circle to trace out the ellipse. Write down several pairs of lengthns as displayed on the screen.
13. Do you notice a relationship between them? (Hint: it has to do with basic arithmetic like +, -, *, and /)
14. Use the measure --> calculate command to display the relationship on screen. Animate to be sure your relationship holds.
15. Be prepared to explain this relationship to your classmates!

Hyperbolas

1. Open the hyperbola.gsp file in Geometer's Sketchpad. Click the animate button. What do you see?
2. What criteria did we use to fold the hyperbola in the previous activity? Is this a definition? How can we prove the hyperbola on the screen follows the same rule?
3. Create a segment from the point that traces the hyperbola to the moving point on the circle.
4. Create another segment from the point inside the circle to the point that traces the hyperbola.
5. Construct a third segment that makes a triangle. Select this segment and construct its midpoint.
6. Construct a segment from the point that traces the hyperbola to the midpoint.
7. Using what you know about basic geometry, can you prove that you are really making a hyperbola? Look back at the definition in number 2, and animate the drawing again if necessary.
8. Hide all of the segments you just made, and the midpoint.
9. We will now call the center of the circle and the point that is inside the circle the foci of the hyperbola.
10. Construct two segements that represent the distance from the point that traces the hyperbola to each of the foci.
11. Use the measure --> length command to display the lengths of each of these segments on the screen.
12. Move the point on the circle around the circle to trace out the hyperbola. Write down several pairs of lengthns as displayed on the screen.
13. Do you notice a relationship between them? (Hint: it has to do with basic arithmetic like +, -, *, and /)
14. Use the measure --> calculate command to display the relationship on screen. Animate to be sure your relationship holds.
15. Be prepared to explain this relationship to your classmates!