GeoGebra Worksheet: TRISECTING A SEGMENT

This is the standard straightedge and compass technique for trisecting any line segment.  We will need the macro that we will create later in another worksheet.  Start with three points, A, B, and C.  We will construct the points that divide AB into three congruent segments.

  1. Uncheck VIEW-AXES.

  2. Select OPTIONS-LABELING-NEW POINTS ONLY.

  3. Construct the ray, r(AC, emanating A and passing through the point C.  Be sure that it passes through C and not just close to C.

  4. Use the "Circle with center and point" tool construct a circle with center A and intersecting r(AC) at a point D: 
  5. Construct the circle with center D and passing through the point A.  This circle will have the same radius as the previous circle.

  6. Find the intersection of the ray r(AC) with this circle, label the point F.
  7. Construct the circle with center F and passing through the point D. Use H to label the other intersection of this circle with r(AC).
     
  8. Construct the segment between B and H, BH.
  9. Construct the line parallel to BH passing through F.
  10. Construct the line parallel to BH passing through D.
  11. Construct the points of intersection of these lines with the segment AB.
  12. Hide all lines, segments and circles.

  13. Construct a tool, called Trisect, which has the three points A, B, and C as initial objects; and has the two trisection points as the final objects.

             

     

     

     

A Triangle within a Triangle

  1. In this investigation we will investigate the area relationship between a triangle and a triangle within it made by connecting each vertex with a trisection point on the opposite side.
  2. Construct a triangle ΔABC using the Polygon Tool.
  3. Use your trisect tool on the points A, B, and C (in that order), to trisect AB.  Execute it again on B, C, and A and C, A, and B to trisect BC and AC.
  4. Going clockwise around the triangle from A to B to C, construct a segment from A to the first point clockwise past B, a segment from B to the first point clockwise from C, and a segment from C to the first point clockwise from A.
  5. Construct the points of intersection of these segments.  Label these points D, E, and F.
  6. Hide the segments and construct the triangle ΔDEF using the Polygon Tool.
  7. What is the relationship between the areas of the inner triangle and the larger outer triangle.  Can you guess what the ratio of these areas is?  Can you compute this ratio? Find the area of triangle ΔABC in your list (possibly poly1) and area of triangle ΔDEF (possibly in poly2) in your list of Dependent Objects.
  8. In the Input window at the bottom of the window, enter Ratio = poly1/poly2.  Find Ratio in your list of Dependent Objects. 
  9. Name the file yourname_04.ggb and email it to me by noon, Wednesday, September 8.