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					                   Macros for regular polygons
                          Cabri figure

Provide constructions for regular polygons and their macros.

Equilateral triangle given a segment.
  1. Construct segment AB, from left to right.
  2. Construct a circle centered at point A with radius to point B.
  3. Construct another circle centered at point B with radius to point A.
  4. Find the point of intersection of the two circles. Label it point C.
  5. Connect point C with points A and B.

                                         C




                            A                          B




  6. To define the macro, use segment AB as the initial object, hide the
     circles, and use segments BC and CA as final objects.
  7. If you draw the original segment from right to left and apply the
     macro, the final triangle will be below the original segment. Measuring
     angles and sides, we can verify that it is an equilateral triangle.

                                    4.11 cm

                                60.0 °            60.0 °

                       4.11 cm                      4.11 cm
                                         60.0 °
Equilateral triangle given a circle.
Steps:
   1. Construct a circle
   2. Draw a radius on the circle.
   3. At the point of intersection of the radius and the circle, draw another
       circle with the same radius.
   4. The two circles are going to intersect in two points, draw a segment
       connecting the two points. This segment is one of the segments of the
       equilateral triangle.
   5. Draw a line over the first radius. The line will intercept the circle. The
       point of intersection is the third vertex of the triangle.
   6. Draw segments that connect the points and make the triangle.




                                          60.0 °
                           5.20 cm                  5.20 cm




                                60.0 °             60.0 °
                                         5.20 cm




   7. Hide one of the circles, the line and the radius.
   8. To define the macro, the circle is the initial object and the three
      segments are the final objects.



Square given segment.
   1. Construct segment AB.
   2. On point A, construct a perpendicular line to segment AB. Do the same
       thing on point B.
   3. Construct a circle centered at point A with radius AB.
4. Find the point of intersection of the circle and the perpendicular line.
   Label the point D.
5. Construct a perpendicular line to line AD through point D.
6. Find the point of intersection between this new line and the



                                  D                    C




                                  A                    B




   perpendicular line to AB through point B. Label the point C.

7. Hide all the lines and circle. Connect the points with segments in this
   order: from B to C, from C to D, and from D to A.
8. Make a polygon over the points, using the polygon tool.


                     D                      C




                     A                      B

9. To define the macro, use segment AB as the initial object and
   segments BC, CD, DA, and the polygon as the final objects.
   10. To verify that the macro makes a square, we can draw a segment, use
       the macro, and then measure the segments and the angles.


                                  3.11 cm
                               90.0 °   90.0 °

                    3.11 cm                      3.11 cm

                               90.0 °    90.0 °
                                   3.11 cm




Square inscribed in a circle

Steps:
   1. Draw a circle.
   2. Draw a line that goes through the center of the circle.
   3. Construct a perpendicular line to the first one, which also goes
      through the center of the circle.
   4. The two lines will intersect the circle in four points that will be the
      vertices of the square.
   5. To define the macro, the initial object will be the circle and the final
      objects are the segments of the square. Hide the lines before
      selecting the final objects.

Pentagon inscribed in a circle.

Steps:
   1. Construct a circle O.
   2. Draw a horizontal radius OB. Construct line AB over the radius.
   3. Construct a perpendicular line, CD, going through the center of the
      circle
   4. Find the midpoint of segment OB. Label the point M.
   5. Using M as the center, construct a circle with radius MC.
   6. Find the point of intersection of the new circle and line AB. Label it N.
   7. Draw segment NC. This length of this segment is the same as the
      length of the sides of the pentagon


                                    C




                   A       N        O      M        B




                                    D




   8. Hide point M and the circle centered at point M.
   9. Construct a circle at point C with radius NC. The points of
      intersection will be vertices of the pentagon (2 and 5).
                                      C




                     2
                                                       5


                    A       N                              B




                                               4

                                     D



Using point 2 as the center, construct a circle with radius NC. Find the point
of intersection of the new circle with the original circle (point 3). Construct
one more circle centered at point 3 with radius NC. Find the point of
intersection 4.



                                                   C




                                2
                                                                   5


                                A      N           O                   B




                                      3                        4

                                                   D
Hide everything, except the original circle and the points C, and 2 to 5.
Using point C as the first point, connect the points 1 to 5. The result is a
pentagon inscribed in a circle.

                                           C




                         2
                                                             5


                                           O




                                 3                   4




   To define the macro, use the original circle as the initial object and the
   five red segments as the final objects. Hide the circle before selecting
   final objects.
   To verify that the macro works, construct a circle, and use the macro.
   Measuring segments and interior angle, we can see that the resulting
   figure is a pentagon.



                        3.43 cm                           3.43 cm
                                          108.0 °


                                108.0 °                  108.0 °


                      3.43 cm                                3.43 cm

                                     108.0 °   108.0 °
                                           3.43 cm
Pentagon given a segment.

Steps:
   1. Given segment AB, find the midpoint M.
   2. Extend line AB and draw a perpendicular line through point A.
   3. Draw a circle at A with radius AB (purple). This circle intersects the
      perpendicular line at C.
   4. At point M, draw a circle with radius MC (yellow). This circle
      intersects line AB at point D.
                                  C




                                  A        M        B          D




   5. Draw another circle at point A with radius AD (red)
   6. Draw a segment from A to D.
   7. Using the compass tool, draw a circle at point B using the segment Ad
       as the radius (red). These two circles intersect at point E.
   8. Draw a circle at point B with radius BA (purple). The two red circles
       intersect the purple circles at points F and G.
   9. Points ABGEF are points of a regular pentagon. Connect the points
       with segments.
   10. Measure angles and sides to make sure it is a regular pentagon.

To define the macro, select segment AB as the initial object. Hide all
unnecessary circles, points and lines. Select the other 4 segments of the
pentagon as the final objects.
                                                E


                                               108.0 °
                                     5.08 cm        5.08 cm
                                 C
                       F                                                   G
                           108.0 °                               108.0 °



                                                                    5.08 cm
                      5.08 cm


                                  108.0 °              108.0 °
                                 A    5.08 cm M               B                D




Hexagon inscribed in a circle.

Steps:
   1. Given circle 0, draw a radius.
   2. The radius will define a point in the circle, 1.
   3. Using the compass and the radius as the new radius, draw a circle at
      point 1.
   4. This new circle will intersect the original one at point 2 and 6.
   5. Using the compass again, draw a circle at point 2 using the same
      radius.
   6. Go around the original circle making new circles until you find all 6
      points that will be the vertices of the hexagon.
   7. Connect the points with segments.

To define the macro, select the first circle as the initial object. Hide all
other circles and unnecessary segments. Select all 6 segments of the
hexagon as the final objects.
                             3                  2




                     4               O              1




                             5              6




Hexagon given segment.

Steps:
   1. Given segment AB, make a circle at A with radius AB and another at B
       with the same radius.
   2. The intersection of the two circles will be point O.
   3. Make a circle centered at point O with radius OA.
   4. This new circle will intersect the other two circles at points C and F.
   5. Using the compass tool, and the segment AB as the radius, make a
       circle at point C.
   6. This last circle will intersect circle O at point D.
   7. Use the compass again to make a circle at point D with radius AB.
   8. The circle will intersect circle O at point E.
   9. Points ABCDEF will be the points of the regular hexagon.
   10. Connect the points with segments.

To define the macro, use the segment AB as the initial object, Hide all other
circles. Use the other segments of the hexagon as the final objects.
                          E
                                       D




                  F                        C
                              O




                          A            B




Regular octagon given a segment.

Steps:
   1. Given segment AB, construct a circle at point B with radius AB.
   2. Draw line AB. It will intercept the circle at point C
   3. Construct the perpendicular to line AB through point B. It will
      intercept the circle at point D.
   4. Bisect the angle CBA. The angle bisector will intercept the circle at
      point E.
   5. Connect points AE with a segment.
   6. Draw perpendicular lines to segment AE through points A, E and B.


                                   D

                                               E




                      A            B               C
7. Draw two points, one at each perpendicular line through point A and E.
8. Bisect the angles XAE and YEA.
9. These two lines will intersect at point F.
10. Draw a circle at point F with radius FA. This is the circle where the
    octagon will be inscribed.




                                                    Y
                                      F
                                            D

                             X
                                                            E




                                 A          B                   C




11. Draw a parallel line to AE through point F.
12. The point of intersection of this line with the circle will give 2 more
    points of the octagon.
13. Connect all the points to get the octagon.
                         I             H




             J                                  G
                                           Y
                                 F
                                      D

                     X
                                                E
                 K




                         A            B                 C
To define the macro, select segment AB as the initial object. Hide all
unnecessary circles and lines. Select the other segments as the final
objects.



Regular dodecagon given a segment.

Steps:
   1. Given segment AB, make a circle at A with radius AB and another at B
       with the same radius.
   2. Draw line AB.
   3. Since the interior angles of a dodecagon measure 150 degrees, Draw a
       perpendicular line to AB at point B.
   4. This line will intercept circle B at point C.
   5. Draw a circle at C with radius CB.
   6. The two circles will intercept at point D
   7. Draw segment AD.
   8. Draw circle at A with radius AD.
   9. The last circle will intercept circle C at point E.
   10. Draw a circle at point E with radius EA. This will be the circle where
       the dodecagon will be inscribed.




                                 E
                                     C

                                             D

                            A            B
  11. Hide all other circles except circle E, A, and B.




                                E



                                               D

                           A         B




  12. Now using the compass tool and segment AB as the radius, make a
      circle at point D. Continue making more circles with the same radius at
      the last point of intersection with circle E.
  13. Connect all the points to get the dodecagon.
  To define the macro, the initial object is the segment AB and the final
  objects are the other eleven segments.




                                      E



                                                   D

                                 A         B




Go back to table of contents.
Go forward to problem 4.1 – Regular Tessellations.

				
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posted:7/13/2011
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