Properties of geometrical figures

Properties of Geometrical Figures

Created by Jade Wright, Prue Tinsey, Tania Young, Garth Lo

Bello and Andrew Roberts

Constructing Geometrical

Figures using GeoGebra

Mathematics • Stage 4

OutcomesA 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

Measurement and GeometryProperties of Geometrical Figures 1

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

Students:

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) Opposite sides are parallel(2) Opposite angles are equal(3) Opposite sides are equal in length(4) Diagonals bisect each other(5) All four angles are right angles(6) Adjacent sides are perpendicular(7) Diagonals are equal in length(8) 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

GeoGebraNow use the tools in GeoGebra to

construct an equilateral triangleIs 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. Create segment AB.2. Construct a circle with centre A through B. 3. Construct a circle with centre B through A.4. Intersect both circles to get point C.5. Create the polygon ABC in counter-clockwise direction.6. Hide the two circles.7. Show the interior angles of the triangle.8. 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

GeoGebraNow use the tools in GeoGebra to

construct an isosceles triangleIs 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. Create segment AB.2. Find midpoint of AB to create point C. 3. Construct a perpendicular line through point C.4. Create new point D on perpendicular line .5. Create the polygon ABD in counter-clockwise direction.6. Hide any lines outside the triangle7. Show the interior angles of the triangle.8. Apply the drag test to check if the construction is correct.

Let’s Look at the Properties of a Parallelogram (1) Opposite sides are parallel

(2) Opposite angles are equal(3) Opposite sides are equal in length(4) Diagonals bisect each other(5) Diagonals do not bisect each other at right angles(6) All angles are not right angles

Now let’s construct a Parallelogram in

GeoGebraNow use the tools in GeoGebra to

construct a parallelogramIs 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. Create segment AB.2. Create new point C not on segment AB.3. Create line parallel to segment AB through point C. 4. Create line BC.5. Create line parallel to segment BC through point A.6. Intersect two lines to create new point D.7. Create polygon ABCD.8. Hide any lines outside the parallelogram.9. Apply the drag test to check if the construction is correct.

Let’s Look at the Properties of a Rhombus

(1) All sides are equal in length(2) Opposite sides are parallel(3) Opposite angles are equal(4) Diagonals bisect each other(5) Diagonals bisect each other at right angles(6) Diagonals bisect angles(7) 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 ActivityNow let’s test your

knowledge1. In pairs students will work together to test each others knowledge.

2. One student will pick a geometrical figure (without telling their partner) and read its properties to the other student.

3. This student will then construct the figure in GeoGebra based on the description given by their partner.

4. 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.

5. Now you will take it in turns until you’ve each constructed each shape.

There are many shapes in the ‘Real World’ that could be modelled using the tools we have just discovered in Geogebra.

An activity that students could do is:1. Download an object using ‘google images’(eg a bridge or a building, and place it on a blank page in Geogebra.2. Use the tools we have just learnt about in Geogebra to create an outline of the shape.3. Once you have finished the teacher will view your work and critique it.

How could this be used in the ‘Real World’?

Thankyou for listening

Any Questions?