Download Properties of Geometrical Figures

Constructing
Geometrical Figures
using GeoGebra
Created by Jade Wright, Prue Tinsey,
Tania Young, Garth Lo Bello and
Andrew Roberts
Measurement and Geometry
Properties of Geometrical Figures 1
Outcomes
A student:
• Communicates and connects mathematical ideas
using appropriate terminology, diagrams and
symbols MA4-1WM
• Applies appropriate mathematical techniques to solve
problems MA4-2WM
• Recognises and explains mathematical relationships
using reasoning
MA4-3WM
• Identifies and uses angle relationships, including
those related to transversals on sets of parallel lines
MA4-18MG
Classify triangles according to their side and angle properties and
describe quadrilaterals (ACMMG165)
 Investigate the properties of special quadrilaterals (trapeziums,
kites, parallelograms, rectangles, squares and rhombuses).
Properties to be investigated include:
― The opposite sides are parallel
― The opposite sides are equal
― The adjacent sides are perpendicular
― The opposite angles are equal
― The diagonals are equal
― The diagonals bisect each other
― The diagonals bisect each other at right angles
― The diagonals bisect the angles of the quadrilateral
 Use techniques such as paper folding, measurement or
dynamic geometry software to investigate the properties of
quadrilaterals (Problem Solving, Reasoning)
 Sketch and label quadrilaterals from a worded or verbal
description (Communicating)
 Classify special quadrilaterals on the basis of their properties
 Describe a quadrilateral in sufficient detail for it to be sketched
Student Activity
In pairs students will use GeoGebra
to construct a variety of geometrical
figures to explore and investigate
their properties
Geometrical Figures
The geometrical figures we are going to
investigate are:
• Rectangle
• Square
• Triangle – Equilateral, Isosceles
• Parallelogram
• Rhombus
Let’s Look at the
Properties of a Rectangle
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Opposite sides are parallel
Opposite angles are equal
Opposite sides are equal in length
Diagonals bisect each other
All four angles are right angles
Adjacent sides are perpendicular
Diagonals are equal in length
Diagonals do not bisect each other at right angles
Now let’s construct a rectangle
in GeoGebra
To open GeoGebra:
• Go to: www.geogebra.org
• Click Download
• Click Applet Start
Now use the tools in GeoGebra to
construct a rectangle
Is your rectangle a construction or just a
drawing?
Use the drag test to determine if it is a
construction
Here’s how to construct a
Rectangle
1. Create segment AB.
2. Create a perpendicular line to segment AB through point B.
3. Insert a new point C on the perpendicular line.
4. Construct a parallel line to segment AB through point C.
5. Create a perpendicular line to segment AB through point A.
6. Construct intersection point D.
7. Create the polygon ABCD.
8. Hide the lines outside the rectangle.
9. Apply the drag test to check if the construction is correct.
Let’s Look at the
Properties of a Square
(1) Opposite sides are parallel
(2) All sides are equal in length
(3) All angles are equal
(4) All angles are right angles
(5) Adjacent sides are perpendicular
(6) Diagonals bisect each other
(7) Diagonals are equal in length
(8) Diagonals bisect each other at
right angles
(9) Diagonals bisect the angles
Now let’s construct a Square in
GeoGebra
Now use the tools in GeoGebra to
construct a square
Is your rectangle a construction or just a
drawing?
Use the drag test to determine if it is a
construction
Here’s how to construct a
Square
1. Create segment AB.
2. Create a perpendicular line to segment AB through point A.
3. Construct a circle with centre A through B.
4. Construct intersection point C.
5. Construct a parallel line to segment AB through point C.
6. Create a perpendicular line to segment AB through point B.
7. Construct intersection point D.
8. Create the polygon ABCD.
9. Apply the drag test to check if the construction is correct.
Let’s Look at the Properties of an
Equilateral Triangle
(1) All three sides are equal in length
(2) All three angles are equal
(3) All three angles equal 60°
Now let’s construct an
Equilateral Triangle in
GeoGebra
Now use the tools in GeoGebra to
construct an equilateral triangle
Is your rectangle a construction or just a
drawing?
Use the drag test to determine if it is a
construction
Here’s how to construct an
Equilateral Triangle
1.
2.
3.
4.
5.
6.
7.
8.
Create segment AB.
Construct a circle with centre A through B.
Construct a circle with centre B through A.
Intersect both circles to get point C.
Create the polygon ABC in counter-clockwise direction.
Hide the two circles.
Show the interior angles of the triangle.
Apply the drag test to check if the construction is correct.
Let’s Look at the Properties of an
Isosceles Triangle
(1) Two adjacent sides are equal in
length called the legs
(2) The angles opposite each of
the equal sides are equal
(3) The other side is called the
base and is not equal in length to
the other two sides
(4) The angle opposite the base is
not equal to the other two angles
Now let’s construct an
Isosceles Triangle in GeoGebra
Now use the tools in GeoGebra to
construct an isosceles triangle
Is your rectangle a construction or just a
drawing?
Use the drag test to determine if it is a
construction
Here’s how to construct an
Isosceles Triangle
1.
2.
3.
4.
5.
6.
7.
8.
Create segment AB.
Find midpoint of AB to create point C.
Construct a perpendicular line through point C.
Create new point D on perpendicular line .
Create the polygon ABD in counter-clockwise direction.
Hide any lines outside the triangle
Show the interior angles of the triangle.
Apply the drag test to check if the construction is correct.
Let’s Look at the
Properties of a Parallelogram
(1)
(2)
(3)
(4)
(5)
(6)
Opposite sides are parallel
Opposite angles are equal
Opposite sides are equal in length
Diagonals bisect each other
Diagonals do not bisect each other at right angles
All angles are not right angles
Now let’s construct a
Parallelogram in GeoGebra
Now use the tools in GeoGebra to
construct a parallelogram
Is your rectangle a construction or just a
drawing?
Use the drag test to determine if it is a
construction
Here’s how to construct a
Parallelogram
1.
2.
3.
4.
5.
6.
7.
8.
9.
Create segment AB.
Create new point C not on segment AB.
Create line parallel to segment AB through point C.
Create line BC.
Create line parallel to segment BC through point A.
Intersect two lines to create new point D.
Create polygon ABCD.
Hide any lines outside the parallelogram.
Apply the drag test to check if the construction is correct.
Let’s Look at the
Properties of a Rhombus
(1)
(2)
(3)
(4)
(5)
(6)
(7)
All sides are equal in length
Opposite sides are parallel
Opposite angles are equal
Diagonals bisect each other
Diagonals bisect each other at right angles
Diagonals bisect angles
All angles are not right angles
Now let’s construct a
Rhombus in GeoGebra
Now use the tools in GeoGebra to
construct a rhombus
Is your rectangle a construction or just a
drawing?
Use the drag test to determine if it is a
construction
Here’s how to construct a
Rhombus
1. Create segment AB.
2. Construct a circle with centre A through B.
3. Construct a circle with centre B through A.
4. Intersect two circles to create new point C.
5. Create line parallel to segment AB through point C.
6. Create line AC.
7. Create line parallel to segment AC through point B.
8. Intersect two lines to create new point E.
9. Create polygon ABEC.
10.Hide any lines outside the rhombus.
11.Apply the drag test to check if the construction is correct.
Student Activity
Now let’s test your knowledge
1.
2.
3.
4.
5.
In pairs students will work together to test each others
knowledge.
One student will pick a geometrical figure (without telling their
partner) and read its properties to the other student.
This student will then construct the figure in GeoGebra based
on the description given by their partner.
The student reading the properties will then check whether
their partner has constructed the correct figure and use the
drag test to test whether it is a “real” construction.
Now you will take it in turns until you’ve each constructed
each shape.
Student Activity
Real World Application
Geometry is used everywhere. Architecture is one example
of how geometry is used in the real world. This task
requires you to investigate the architecture used to design
the Eiffel tower
The Task
In groups of 2 examine the geometry that can be found in
architecture used to build the Eiffel tower. You will need to:
1. Label the geometric shapes in the Eiffel tower, using
images found from internet resources. You can investigate
these shapes in GeoGebra to verify their properties.
2. Create a PowerPoint presentation to report these
findings. Concepts that you should cover, include:
- a brief history on the time and place of construction
- why it was constructed
- a description of your geometric findings, regarding the
architecture of the building. For example, the shapes
used to create the building, and an explanation why this
geometry is so important in the building of the Eiffel
tower.
Thankyou for listening
Any Questions?