Workshop on 3D Dynamic Geometry: Cabri 3D
Assessment and HKEAA Section,
Education Bureau.
Hong Kong Education City
Jun 2009
To download a free 30-day full version of Cabri 3D V 2.1.2, visit www.cabri.com.
1.
The Cabri 3D Environment
Toolboxes
Origin and i, j, k
vectors
Base plane (visible
part)
Right-button drag to change the view angle.
The toolboxes:
Manipulation
Lines and
Relative
Regular
Regular
& Redefine
Curves
Constructions
Polygons
Polyhedra
Points
Planes and
Transformations
Polyhedra
Surfaces
Each toolbox contains a number of tools. For example, here is the curves
toolbox, with the vector tool selected.
To use a tool, click on the toolbox, or click and hold, and drag to the required
tool.
P.1
Measurements
2.
Creating, manipulating and labeling points
1.
Use the “Point” tool
2.
3.
Create a point on the base plane.
Label it as ‘A’ by pressing A.
4.
5.
6.
Press ‘Shift’ and move the mouse, click to
create a point in the space.
Label this point as ‘B’.
Drag to move it horizontally.
7.
Move it vertically by pressing ‘Shift’ and drag.
.
3. Creating a XYZ box
1. Use the “XYZ Box” tool.
2. Click on points A and B to create a XYZ
box.
3. Drag A or B, or press Shift and drag B, to
see how the size of the box change.
4. Drag the box to reposition the box.
5. Press and hold “Ctrl” and click on points A and B,
right-click and choose “Hide/Show”, or press
6.
Ctrl-M.
Ctrl-click the labels A and B and press the “Del”
button.
7.
Select the origin and i, j and k vectors and hide them.
4.
Two largest cubes in a box
Problem: Two identical cubes are put inside a box of dimensions 8cm ×
1.
2.
3.
4.
6cm × 5cm. What is the largest possible volume of each cube?
Right-click on the surface of the XYZ box, choose “Surface Style |
Empty” to empty the faces of the box.
Use the “Length” tool, click the segment QR to measure its length.
Click on a length, a width and a height to measure the dimensions of
the box.
Drag the vertices horizontal and
vertically (shift-drag) so that the
dimensions become 8cm × 6cm ×
5cm.
Use the “Cube” tool, click on the
plane, then click on two positions on the plane
to create a cube centred at the first position and
with the second position as a corner. Click on
P.2
5.
one vertical face to create another identical cube.
Drag any vertex to enlarge and rotate the cubes. Drag the top
face to shift the cubes. When will the two cubes be largest?
5.
1.
2.
3.
Embedding in PowerPoint
Right-click on the empty space and choose “Copy”.
Open an empty PowerPoint, right-click and choose “Paste”.
Project the slide, right-click on the figure. Drag and right-button drag to see
6.
1.
Counting vertices, edges and faces of a regular dodecahedron
Create a regular dodecahedron using the
“Regular Dodecahedron” tool.
Right-click the dodecahedron, choose “Point
Size | Empty” to empty the vertices.
Use the “Point” tool to click on each vertex to
count the number of vertices (V).
Use the “Segment” tool, click on each edge to
2.
3.
4.
5.
6.
7.
1.
2.
3.
count the number of edges (E).
Use the “Polygon” tool, click on each face to
count the number of faces (F).
Verify Euler’s formula “V – E + F = 2”.
Dual of the cube
Create a cube. Empty its faces.
Use the “Midpoint” tool. In each face, click on
any two opposite vertices to create the centre of
the face. Choose the “Biggest” size for the
centres.
Use the “Convex Polyhedron” tool. Click the
centres of the faces, and double-click the last point
to obtain the dual of the cube.
Step 2
Step 3
P.3
8.
Cross-Sections of the Cube
1.
Create a cube. Use the “Point” tool
. On each of three adjacent faces, create a point of
biggest size.
2.
Use the “Plane” tool
, click these three points to create a plane passing through them.
3.
Use the “Cut Polyhedron” tool, click the cube
and the plane to cut the cube by the plane.
4.
Press F8, or choose Windows | Active View, to
show the Active View window. Click the “Show
Hidden Object” checkbox. Show the hidden cube by right-click it and choose “Hide/Show”, or
press Ctrl-M. Uncheck the “Show Hidden Object” checkbox.
5.
Use the “Polygon” tool
6.
“Surface Color” to change its color to yellow.
Empty the face of the cube. Hide the plane. Drag the three points to see cross-section of various
, click on the cross-section. Right-click the cross-section, choose
shapes.
Step 1
Step3 – 4
Step 5 – 6
2.
A movable segment in the XYZ box and its projection
Create a XYZ box. Empty its faces. Use the Point tool to create the vertices. Label the box as
‘ABCDEFGH’ as shown.
Use the “Segment” tool to create sides BC and EF.
3.
Step2
Step 3 – 9
Step 10
Use the “Point” tool to create a point on each of BC and EF. Label them as P and Q respectively.
4.
Use the “Perpendicular” tool
9.
1.
5.
, click on P and then the base to draw a perpendicular line from
P to the base.
Use the “Intersection Point(s)” tool, click on the
perpendicular and the base plane to find the
intersection of the perpendicular and the base.
P.4
6.
7.
8.
9.
Label the intersection as ‘R’.
Click on and hold in the empty space to see the movable points.
Select P and Q. Right-click and choose “Point Size | Biggest”.
Use the “Triangle” tool to draw the triangle PQR.
Right-click on it and choose “Surface Color” to
yellow.
10. Select the perpendicular and the two segments
and hide them. Move P and Q to see.
10. Measuring Angles and Lengths
1. Use the “Angle” tool, click P, Q and R
2.
3.
4.
respectively to measure ∠PQR.
Measure the right-angle ∠PRQ using the
“Angle” tool. Hide the measurement
“90°”.
Use the “Length” tool, click the segment QR to
measure its length.
Drag Q to see the variations of the angles and the
projection.
S
Step 1 – 3
11. Angle between 2 planes in the cube
1. Create a cube ABCDEFGH of empty faces as shown.
2.
Use the “Triangle” tool to create ΔAFH and ΔAGH.
Step 2
Step 3 – 7
Step 8 – 10
3. Use the “Segment” tool to join AH.
4. Create a movable point on it, label it as ‘P’.
5. Create segments AF, AG, FH, GH.
6. Use the “Perpendicular” tool to create a plane at P perpendicular to AF.
7. Find the intersections between the plane and the segments in Step 5.
8. Hide unwanted objects.
9. Join P and the intersections by segments.
10. Create a yellow triangle with vertices being P and the other two intersections
11. Measure the angle between the two planes. What is this angle? Why?
P.5
12. Constructing a prism and its cross sections
1. Use the “Equilateral Triangle” tool
to create an equilateral triangle.
2.
3.
4.
5.
6.
7.
8.
Construct a perpendicular through one of the
vertex. Create a movable point on it.
Use the “Vector” tool to construct a vector
from the vertex to the movable point.
Using the “Prism” tool, click on the vector and
then the triangle to create a prism.
Change the surface style of the prism to “Small
Hatches”.
Create a movable point P on the vector.
Create the other 2 vertical edges of the prism
using the “Segment” tool.
Use the “Perpendicular” tool
to create a plane through P and perpendicular to the vector.
9. Find the intersections of the plane and the 2 other vertical edges. Hide the plane.
10. Use the “Triangle” tool to draw the triangle whose vertices are P and the 2 intersections.
11. Hide the unwanted objects as shown. Drag P to see the cross section at different height.
Step 1 – 2
Step 3
Step 4
Step 5 – 8
Step 9 – 10
P.6
Step 11
13.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Redefining a point, constructing a pyramid and its cross sections
Hide the origin and i, j, k vectors.
Use the “XYZ Box” tool to create a rectangle on the base plane.
Create a point in the space.
Using the “Convex Polyhedron” tool, click on this point and the rectangle to create a pyramid.
Drag the apex to see.
Change the view to see the bottom of the base plane. Construct the centre of the base rectangle.
Construct the perpendicular through the midpoint.
Use the “Redefinition” function. Click on
the apex and then the line to redefine the
apex as a point on the perpendicular. Drag
the apex to see.
Change the surface of the pyramid to “Small Hatches”.
Create the 4 slant edges using the “Segment” tool.
Construct a movable point on one of the edge.
Construct a plane through this point perpendicular to the altitude.
Find the intersections of the plane and the other 3 slant heights.
Use the “Polygon” tool to draw the cross section.
Hide all unwanted objects as shown.
Step 1 – 3
Step 4
Step 6 – 7
Step 8
Step 9 – 13
Step 14 – 15
P.7
14. Rotational symmetry of a cube
1. Create a cube. Create the centres and the mid-points of the edges of the top and the bottom.
Choose the colour of the mid-points as blue. Hide the base plane.
2. On each of the top and the bottom, create a movable point of biggest size. Join them using the
“Line” tool
3.
4.
5.
6.
7.
.
Create a movable point on the line. Construct a plane through this point perpendicular to the line.
Use the “Circle” tool to construct a circle
around the line and passing through a point on
the plane. Hide the plane. Join the point to the
centre by a segment.
Construct a movable point ‘P’ on the circle.
Change its size to biggest. Join it to the centre.
Use the “Rotation” tool, click the line, then the
point on the circle and then P, to rotate the
cube about the line from the point to P.
Empty the surface of the fixed cube. Hide the other unwanted objects. Adjust the axis and rotate
the cube to find the rotational symmetry.
Step 1 – 2
Step 3 – 4
15. A net of the regular tetrahedron
1. Use the “Regular Tetrahedron” tool
2.
3.
4.
5.
6.
Step 5 – 7
to create a regular tetrahedron.
Use “open polyhedron” tool, click on the
tetrahedron to create a net.
Drag any surface to fold or unfold the net.
Press “F8”, click “Show Hidden Objects”
to see the hidden tetrahedron.
Click the tetrahedron, press “Ctrl-M” or
right-click and choose “Hide/Show” to show the tetrahedron.
Close the Active View Window. Empty the surface of the tetrahedron.
Step 1 – 2
Step 3
Step 4
P.8
Step 5 – 7
Advanced Tasks
16. A paper aeroplane (HKCEE 99)
1. Draw an equilateral triangle
2. Draw a circle thro’ the midpoint around the
with the median and a midpoint.
segment. Draw the perpendicular.
4. Create this triangle and rotate
it about the median from the
midpoint to the movable point.
3. Construct this quadrant using the “Arc”
tool. Create a movable point on it.
about the segment
to
5. Construct a line thro’ the vertex //
to the edge of the triangle.
from
7. Draw this segment.
6. Construct the plane
thro’ the movable point
and the line.
8. Draw the circle around the segment
thro’ the vertex. Hide the triangle.
P.9
9. Draw the triangle using the
segment and the intersection between
the circle and the plane.
11. Hide unwanted objects.
10. Draw this triangle.
12. Construct the plane thro’ the
perpendicular and the vertex.
14. Hide unwanted objects. Drag this
point to fold or unfold.
13. Construct the other
half by reflection.
P.10
17. Another net of the regular tetrahedron
Construct the segment and the vector.
This circle passes thro’ the apex and
is around this segment.
First create a movable point on the plane.
Translate it by the vector and create the
segment.
Construct a movable point on the segment
and the perpendicular plane.
Construct the plane and find its
intersections with the circle.
Create the triangle face and rotate it about
the edge from the apex to the intersection.
Construct a perpendicular thro’ the centre.
about this edge
from
to
Construct the base.
P.11
Construct another perpendicular from
the centre of this triangle
Rotate the base
about
from
to
Rotate this triangle
Hide unwanted objects. Drag up
and down to fold or unfold.
about
from
to
P.12