Cabri II Resource Pack

Cabri II Resource Pack

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University of Cumbria Cabri II Resource Pack

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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


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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.

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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.


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Using Cabri to Explore the Centres of Triangles A

This is a the circumcircle of a triangle:

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Use cabri to create the circumcircle of a triangle.

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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.


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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?


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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?

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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.


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Using Cabri to explore the trigonometrical ratios for general angles Use the diagram below to help you. 1 2 3 4

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.

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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?


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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


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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?


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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)


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Some General Cabri Investigations 1

Pentagon Puzzle

Can you construct a regular pentagon? Your completed pentagon should not ‘come apart’ when pulled.

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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.

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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.


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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.

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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?

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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


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Cabri II Resource Pack

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