Cabri II Resource Pack
PLEASE DO NOT REMOVE
University of Cumbria Cabri II Resource Pack
1
Exploring the Cabri Tools
Activity 1
Explore the features of Cabri. You might like to try to:
Create a circle
Create a line segment
Draw 3 points and the line segments between them (i.e a triangle)
Draw 3 lines and put 3 points upon them (ie another triangle)
Draw a quadrilateral
Draw other polygons
Put angle markers on an angle of one of your polygons and look
what happens when a vertex is ‘dragged’.
Activity 2
Use Cabri to perform the following constructions:
Create an angle of 60º
Bisect an angle
Construct an angle of 30º
Construct a perpendicular to a line
Bisect a line
Construct a right angled triangle with angles 90º, 60º, 30º
You may do these in any order
University of Cumbria Cabri II Resource Pack
2
Using Cabri to explore the properties of triangles
1
Constructing triangles
Use Cabri’s construction tools to construct:





A scalene acute triangle
A scalene obtuse triangle
A right angled triangle
An isosceles triangle
An equilateral triangle
Label all the angles and side lengths. Consider what mathematical skills
and knowledge you needed to complete this task.
2
The angle sum of a triangle.
Use Cabri to find the angle sum of a triangle.
Attempt to use Cabri to construct a proof for the angle sum of a triangle.
You may wish to save and print off your work.
3
The exterior angle of a triangle.
Use Cabri to explore the relationship between the exterior angle of a
triangle and the interior angles.
Attempt to use Cabri to construct the proof for the exterior angle of a
triangle.
You may wish to save and print off your work.
University of Cumbria Cabri II Resource Pack
3
Using Cabri to Explore the Centres of Triangles
A
This is a the circumcircle of a triangle:
1
Use cabri to create the circumcircle of a triangle.
2
It is said that the centre of the circumcircle, which is
called the circumcentre, is equidistant from the 3
vertices of the triangle. Test the veracity of this
statement by constructing the circumcircles of
triangles.
B
This is the incircle of a triangle:
1
Use Cabri to construct the incircle of a triangle.
2
It is said that the centre of the incircle, which is known
as the incentre, is equidistant from the sides of the
triangle. Test the veracity of this statement by
constructing the incircles of triangles.
C
A median of a triangle is defined as a line joining a vertex to the
midpoint of the opposite side.
1
2
3
4
5
D
Investigate the properties of the
medians for a particular triangle.
What do you notice?
Test your ideas for other triangles.
Can you generalise your hypotheses.
Can you prove your hypotheses using
geometric theorems?
The altitude of a triangle is defined as a line drawn from a vertex
perpendicular to the opposite side.
1 Investigate the properties of altitudes for
a range of triangles.
2 Attempt to generalise and prove your
findings.
University of Cumbria Cabri II Resource Pack
4
Using Cabri to Explore the Areas of Triangles
It is said that the area of a triangle can be found using the following
formula:
Area = s(s  a)(s  b)(s  c)
Where a, b, and c are the lengths of the sides of the triangle and s is the
semiperimeter i.e. ½ (a + b + c)
In cabri, select the polygon tool to create a triangle. Use the calculation
tool to find its area.
Does the above rule appear to be true? Test it for several triangles.
Can you explain/prove your findings?
University of Cumbria Cabri II Resource Pack
5
Using Cabri to explore Pythagoras’ Theorem
1
Pythagoras’ Theorem
Use Cabri to construct the squares on the sides of a right angled triangle.
Test the hypothesis:
The square on the hypotenuse is equal to the
sum of the squares on the other two sides.
Test this for several triangles. Does it seem to be true?
2
Exploring Pythagoras’ Theorem further
Above you have tested the theorem. Use Cabri to explore:
What shapes other than squares does the theorem hold for? Record
your findings and produce some printouts to illustrate them.
Can you prove your findings?
3
Extension
Does the theorem only hold for right
angled triangles? Test the squares on
other types of triangles. Record and
attempt to justify your findings.
University of Cumbria Cabri II Resource Pack
6
Using Cabri to explore the trigonometrical ratios for
general angles
Use the diagram below to help you.
1
2
3
4
5
Create a circle. Label the centre O.
Create a point upon the circumference of the circle. Label this point P.
Construct a line segment between the centre of the circle and point P.
Create a point Q perpendicular to the abscissa and the line passing though P.
Measure the angle between the abscissa and the line segment constructed from
the centre of the circle to the point P.
6 Use the “Measure” menu to find the coordinates of O, P and Q.
7 Move point O until its coordinates are (0, 0)
8 Set up calculations to record tan θ using the coordinates of O, P and Q rather
than the distances.
Explore:
What happens to the value of tan θ as the point P moves from the 1st to the 2nd to the
3rd and to the 4th quadrant?
Can you set up calculations to record sin θ and cos θ using cabri?
Using either cabri or pencil and paper method, find and record what happens to the
values of sin θ and cos θ as the point P moves 1st to the 2nd to the 3rd and to the 4th
quadrant?
University of Cumbria Cabri II Resource Pack
7
Using Cabri to explore the properties of quadrilaterals
Using line segments, create a quadrilateral and label it XYZW
Label the point where the diagonals of the quadrilateral intersect O.
Measure the lengths of XO, OW, OY and OZ.
Now change the shape of the quadrilateral. What appears to happen as
the length of XO and OY get closer to the lengths of OW and OZ
respectively?
Can you prove your conjecture?
X
Y
O
W
Z
University of Cumbria Cabri II Resource Pack
8
Using Cabri to explore angles in the same segment
Major segment
chord
Minor segment
P
OO
A
B
We define angle APB as the angle in the segment (in this case the major segment).
Investigate angles the same major segment. What do you notice?
Investigate the angles in the minor segment. What do you notice?
Can you prove any findings?
University of Cumbria Cabri II Resource Pack
9
Using Cabri to explore invariance
Start up Cabri
1. Click on “Show Axes”
2. Click on “Define Grid”
3. Click on the axes to make
the grid appear on the
screen.
4. Create a suitable 2D figure.
Use the Cabri transformation tools to:
Reflect your figure:
In the lines x = 0, y = 0, y = 0, another line.
Explore what happens when you move various
‘parts’ of your object. Think about what you
need to label.
Rotate your figure:
Using various centres of rotation for a given angle e.g. 90º clockwise.
Explore what happens when you change the centre of rotation and/or the angle of
rotation. Think about what you need to label.
Translate your figure by various vectors.
Explore what happens when you change the dimensions of your figure. Think about
what you need to label
You might find the following recording table helpful.
Attribute
Variant (Changes)
Invariant (Doesn’t change)
Number of sides
Length of sides
Size of angles
Colour of shape
Position in 2D space of any point in the original
Orientation of shape
Other attributes (state these)
University of Cumbria Cabri II Resource Pack
10
Some General Cabri Investigations
1
Pentagon Puzzle
Can you construct a regular pentagon? Your completed pentagon should not ‘come
apart’ when pulled.
2
Perpendicular Bisectors
*this is a diagram only –
perpendicular bisectors are
approximate
This is a quadrilateral together with the perpendicular bisectors of the 4 sides. *
Can you find a quadrilateral which has 4 perpendicular bisectors crossing at one
point? Find as many answers as you can.
Try to find out what is special about the quadrilaterals which work, and why.
Explore the perpendicular bisectors of pentagons, hexagons or other polygons.
3
Reflection
 Reflect the triangle in the mirror line.
 What shapes can you make by dragging the triangle?
 What shapes can’t you make?
 Explain your results.
University of Cumbria Cabri II Resource Pack
11
4
Equal Sides, Equal Angles
This pentagon has equal sides but the angles are
not all equal.
This hexagon has equal angles but not equal sides.
Find ways to generate polygons which have equal sides or equal angle but not both.
5
Hexagon hallucinations
You might like to try this investigation on paper first.

Draw an irregular hexagon. Join up the mid-points of the sides. Join up the
mid-points again. Continue the procedure.

What do you notice? Explore this idea using Cabri.

Can you generalise what you ‘see’?

Can you prove what you see using Cabri or anther method?
6
The problem of Apollonius
Given three things, each of which may be a point, a line or a circle, draw a circle that
is tangent to each of the three given things (where tangency to a point is to be
understood to mean that the circle passes through the points).
Hint: There are 10 possible solution cases
University of Cumbria Cabri II Resource Pack
12