Constructing triangles by S9AJ32J8

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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. Save your constructions to file.
Print off at least 1 construction. Consider what mathematical skills and
knowledge you needed to complete this task.

2          The angle sum of a triangle.

Use Cabri to construct the proof for the angle sum of the triangle.

Save and print off your work.

What happens when you ‘drag’ the various points? Consider the
implications of this for teaching and learning in the primary school.

Does this method constitute a formal proof?

3          The exterior angle of a triangle.

Use Cabri to construct the proof for the exterior angle of a triangle.

Save and print off your work.

What happens when you ‘drag’ the various points? Consider the
implications of this for teaching and learning in the primary school.

Does this method constitute a formal proof?

Cabri Investigations FML                                                   1
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?

Consider the implications of using Cabri for exploratory geometry work in
the primary school.

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.

Cabri Investigations FML                                                  2
Right Angled Triangles

Use Cabri to investigate the proprieties
of right angled triangles.

Hints:
 Consider triangles whose side lengths are whole
numbers.

 Consider triangles whose side lengths are not
whole numbers.

 Consider the ratio of the sides.

 You may use additional software packages to
support presentation and analysis of your results.

Cabri Investigations FML                                      3
Using Cabri to explore similarity
Use Cabri to investigate triangles with different scale factors of
enlargement including fractional and negative.   Use a centre of
enlargement outside the figure.

In what way is performing an enlargement on Cabri different to the
standard computer drawing package?

Explore changes in area under enlargement.

What happens when you use Cabri’s dynamic facility to move the centre
of enlargement?

Investigate Cabri’s transformation tools.   Consider their use for
supporting ideas related to transformations. Make notes on your
observations.

Cabri Investigations FML                                            4
Angles in the same segment

Major segment

chord

Minor segment

P

O
O
B
A

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?

Cabri Investigations FML                                                            5
Quadrilaterals in circles investigation
Construct a quadrilateral inside a circle.

Move the vertices around the circumference of the circle. Note what
happens to the angles of the quadrilateral.

What else can you notice as you move the vertices around the
circumference?

Record your findings as a succinct statement.

Can you prove your statement using pencil and paper methods?

Alternate segments investigation
Construct a triangle inside a circle. Draw a line segment to produce a
tangent to the circle which touches one of the vertices of the triangle.

Move the vertices of the triangle around the circumference of the circle
and note what happens to the angles.

Will your observation always be true?

Record your findings as a succinct statement.

Can you prove your statement using pencil and paper methods?

Cabri Investigations FML                                               6
Using Cabri to explore quadrilaterals

Use Cabri to find as many different quadrilaterals as possible with:

 Zero sides of equal length

 2 sides of equal length

 3 sides of equal length

 4 sides of equal length

     No angles of equal size

     2 angles of equal size

     3 angles of equal size

     4 angles of equal size

     Zero pairs of parallel sides

     1 pair of parallel sides

     2 pairs of parallel sides

In each case above, write down the name of the quadrilateral(s) found.

Can you now classify quadrilaterals according to the 3 properties
investigated?

Page 23

Cabri Investigations FML                                                 7
More 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.

Cabri Investigations FML                                                              8
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.

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.

Cabri Investigations FML                                                           9
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 golden ratio
The golden ratio can be defined as:

A division of a line into 2 segments such that the ratio of the larger
segment to the smaller segment is equal to the ratio of the whole line to
the larger segment.

A                          P                  B

AP AB
So AP : PB = AB : AP which is the same as           
PB AP

Use Cabri to investigate if you can construct the golden ratio and hence
calculate the value of the golden ratio (known as Phi Ф).

Cabri Investigations FML                                                10
7       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.

Cabri Investigations FML                          11
7       Centres of Triangles (cont.)

C A median of a triangle is defined as a line joining a vertex to the
midpoint of the opposite side.

1     Investigate the properties of
the medians for a particular
triangle.
2     What do you notice?
3     Test your ideas for other
triangles.
4     Can you generalise your
hypotheses.
5     Can you prove your hypotheses using geometric theorems?

D 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.

Cabri Investigations FML                                                12
8     Arithmetic Composition
The image below is based on a painting by Theo Van Doesbury (1883 – 1931). The
problems based on this image and given below are adapted from: Sharp, J (2004)
Art Inspires Geometry Mathematics in School, January 2004

Please note, the image below has been reproduced in MSWord so will not be exactly
to scale.

How the image is constructed
The composition above is based on drawing a tilted square in the isosceles triangle
formed by bisecting the square whose base is on the diagonal and whose other
vertices are on the sides of the original square. Another horizontal square is created
whose side is half the side of the original one and the process is repeated.
(See next page for investigation questions.)

Cabri Investigations FML                                                            13
8     Arithmetic Composition (cont.)
Starting with a unit square, what is the side of the resultant
square? What proportion is this of the diagonal?

What is the relative area of two successive squares?

Doesburg only paints four squares. Suppose the construction
is carried on infinitely. What proportion of the original square is
the area of the tilted squares?

Consider other geometric compositions and investigate them.

Cabri Investigations FML                                          14

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