Geometric properties of Triangles Quadrilaterals Polygons.

Geometric properties of

TrianglesTriangles

QuadrilateralsQuadrilaterals

PolygonsPolygons

Scope of Presentation

Presented by

Yan Hee Cheong

Lawrence Yong Shao Ping

Damien Yeo Tat Sheng

Tan Ing Keat

Elvin Yeo Boon Heng

The Egyptian Pyramid

The US Pentagon

How would you describe these structures ??

Mathematically, they are structures of very unique

and magnificient ‘Geometric’ designs

This lead us to our topic for today : Geometric properties of Triangles, Quadrilaterals and Polygons

Some History of Geometry

• Egyptians ( 2000 – 500 B.C.)

Ancient Egyptians demonstrated practical knowledge of geometry through surveying and construction of projects

• Babylonians ( 2000 – 500 B.C.)

Ancient clay tablets reveal that the Babylonians knew the Pythagorean relationships.

• Greeks ( 750 – 250 B.C.)

Practiced centuries of experimental geometry.

The greatest mathematical textbook of all time is the Elements – written by Euclid of Alexandria (320 to 260 B.C.). The book had dictated the study of geometry for > 200 years

Euclid

Basic elements of Geometry

URL : Basic elements of geometry

Basic elements of Geometry

Points, Lines and Angles can be manipulated to form various types of geometrical shapes and sizes

Van Hiele Theory of Geometric Thought

1. The Model (5 distinct levels)level 0 – Visualizationlevel 1 – Analysislevel 2 – Informal Deductionlevel 3 – Deductionlevel 4 – Rigor

2. Properties of the Model Sequential

3. Phases of Learning Each level separated by a learning phase

0 Visualization

1 Analysis

2 Informal Deduction

3 Deduction

4 Rigor

Level Description

Identify an object by its appearancePhases of

learning

Formulates and uses definitions

Ability to state proofs

Identify properties of a class of figures

Analyzes various deductive systems

(Upper Secondary)

(Upper Primary)

(Lower Secondary)

(Lower Primary)

(JC, University)

Phases of Learning

Teaching-learning act Teacher stimulate students to learn and

construct meaning in their learning

Phase 1: Inquiry/Information

Phase 2: Directed Orientation

Phase 3: Explication

Phase 4: Free Orientation

Phase 5: Integration

Activity 1

Level 2Informal Deduction: A network of relations begins to form

• You are given a pile of toothpicks all the same size.• First pick three toothpicks.• Can you form a triangle using all three toothpicks

placed end to end in the same plane?• Can a different triangle be formed? • What kinds of triangles are possible? • Now take four toothpicks and repeat the questions.

Then repeat with five toothpicks, and so on.

No. of

toothpicks 3 4 5 6 7

Is triangle possible?

No. of triangles

Kind of triangles

1. Using toothpicks

Y N Y Y Y

1 0 1 1 2

Equilateral Isosceles Equilateral Isosceles

Activity Two : Exploring quadrilaterals

Quadrilaterals are four-sided figures

Activity 2 : Classification of quadrilateralsMatch the following property cards to the following figures

4 sides equal

4 right angles

Opposite sides parallel

Diagonals congruent

opposite sides equal

opposite angles equal

Diagonals not congruent

4 sides equal

4 sides equal

4 right angles 4 right angles

Diagonals congruent Diagonals congruent


opposite sides equal opposite sides equal

opposite angles equal opposite angles equal

opposite angles equal opposite angles equal

Diagonals not congruent

opposite sides parallel opposite sides parallel

opposite sides parallelopposite sides parallel

SQ

UA

RE

RE

CT

AN

GL

E

RH

OM

BU

S

PA

RA

LL

EL

OG

RA

M


Interesting Problems

• What’s the largest rectangle that can be inscribed in an equilateral triangle ?

Hint: The first task is to maximize the dimensions of the inscribed rectangle….

• Stealth Technology makes use of geometrical properties of polygons

• Stealth Technology being used in airplanes, objective being

-To make an airplane invisible to radar waves

Egs of such airplanes are F-117A

(hexagonal shape)

Stealth Technology

F-117A NightHawk (Triangular shape)

Because of their geometrical shapes, they are able to reflect radar signals and thus able to serve its function as a warplane efficiently.

Folding Activities

1) A sheet of paper is rectangular. How can I use folding to make a perfect square without making any measurements?

Explain why your method produces a perfect square…

qp

S R

q

R

p q

S R

z

2)

Step 1: Obtain a square piece of paper as shown in the diagram on the right

Step 2: Fold the square in half, so that PS lies exactly on top of QR. Crease carefully along the middle vertex

Step 3: Fold along the line through R and Z; crease and then unfold. Then fold along the line through S and Z to make a third crease line.

What is special about the triangle RSZ?

Triangle RSZ is an isosceles triangle. Can you verify for yourself?……

(Side ZR = Side ZS)

p q

S R

z

Extension of folding activities….

• Use folding to find the special point Z’ on the centre-fold for which the triangle RSZ’ is an equilateral triangle

Explain why the triangle RSZ’ is an equilateral triangle

S R

Z’

Interactive Geometry

Other interesting problems…

Errors and Misconceptions

• No. In general, N-sided figures are joined by lines and not curves.

• Is there such thing as a 2 sided Figure?


• There is no difference and there is no such thing as an inverted triangle.

• What is the difference between the two figures?


• Yes, this is a four sided figure joined by lines at its end.

• Can this be considered a quadrilateral?


• Is sum of exterior always equals to 360 ?0


• Lets consider a regular convex octagon.

1350 1350

1350

1350

13501350

1350

1350

450

450

450

450

450450

450

450

450 x 8 = 3600• Sum of exterior =

• Sum of exterior

= 1500 + 1500 +1200 + 1200

=5400


1200300 300

2400

600

300 300

• It seems that this is only true for convex figures?

1200

1500

15001200

• If there are reflex angles,

MISCONCEPTION

• Sum of exterior

= 1500 + 1200 +1500 + (-60)0

=3600


1200300 300

2400

600

300 300

• Thus it is also true for convex figures!• Hence it is true for all polygons!

1200

1500

1500

-600

• There must be consistency in measuring exterior angles


• No. it can also have up to 2 parallel sides.YES

NO

• Is it true that the trapezium has only one parallel sides?

• So, is a parallelogram a trapezium?

• Is a trapezium a parallelogram?

Testing your concepts (True or False)

• A square is a rectangle but a rectangle is not a square.

• Squares, rectangles, rhombus, parallelogram and trapeziums are all quadrilaterals. True

True

Grouping Quadrilaterals

• Grouping four sided figures using a Venn diagram

• ε as the universal set for quadrilateralsε

Rhombus

square

Rectangle

Parallelogram

Trapezium

square Rectangle


True

False• A rhombus is a rectangle.

• A square is a rhombus but a rhombus is not a square.



True



• ε as the universal set for quadrilaterals

Rhombus square

ε

Parallelogram

Trapezium

square Rectangle

Rhombus

Rhombus square Rectangle


• A parallelogram is a trapezium but a trapezium is not a parallelogram.

• Squares, rectangles, rhombus are parallelograms True

True

False


• A rhombus is a rectangle.

• A rhombus is a parallelogram but a parallelogram is not a rhombus.

• A square is a rhombus but a rhombus is not a square.


True

True

True

• Squares, rectangles, rhombus are trapeziums True



• ε as the universal set for quadrilateralsε

Rhombus square Rectangle

Parallelogram

TrapeziumParallelogram Trapezium

Questions and Answers

Geometric properties of Triangles Quadrilaterals Polygons.

Documents

congruent slide

learning phase slide

university slide

integration slide

foursided figures slide

years euclid slide

us pentagon slide

egyptian pyramid slide