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GEOMETRY 1

1. 2 3 Optical Illusions through art using geometric concepts 1.1 GEOMETRIC ILLUSIONS

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Page 1: 1. 2 3 Optical Illusions through art using geometric concepts 1.1 GEOMETRIC ILLUSIONS

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GEOMETRY

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GEOMETRY

GEO means “Earth”METRY means “Measure”

GEOMETRY is also about shapes, their properties and relationships

GEOMETRY is about measuring

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Optical Illusions through art using

geometric concepts

1.1GEOMETRIC ILLUSIONS

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Penrose triangleThe Penrose triangle, also known as the Penrose tribar, is an impossible object.

GEOMETRIC ILLUSIONS

The impossible cube or irrational cube is an impossible object that draws upon the ambiguity present in a Necker cube illusion. An impossible cube is usually rendered as a Necker cube in which the edges are apparently solid beams. This apparent solidity gives the impossible cube greater visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. The illusion plays on the human eye’s interpretation of 2-dimensional as 3-dimensionall objects.

1.1

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A famous perceptual illusion in which the brain switches between seeing a young girl and an old woman

GEOMETRIC ILLUSIONS 1.1

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The Kanizsa triangle is an optical illusions first described by the Italian psychologist Gaetano Kanizsa in 1955. In the accompanying figure a white equilateral triangle is perceived, but in fact none is drawn. This effect is known as a subjective or illusory contour. Also, the nonexistent white triangle appears to be brighter than the surrounding area, but in fact it has the same brightness as the background.

GEOMETRIC ILLUSIONS 1.1

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The Necker cube is an ambiguous line drawing. It is a wire-frame drawing of a cube in isometric perspective, which means that parallel edges of the cube are drawn as parallel lines in the picture. When two lines cross, the picture does not show which is in front and which is behind. This makes the picture ambiguous; it can be interpreted two different ways. When a person stares at the picture, it will often seem to flip back and forth between the two valid interpretations (so-called multistable perception).

GEOMETRIC ILLUSIONS

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Congruent means that the figures must have same shape and size

Similar means figures must have same shape, but can also have same size

1.3Identifying Congruent and Similar Figures

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Exploring Symmetry 1.4

A figure has Line Symmetry or Reflection Symmetry if it can be divided into 2 parts, each of which is the mirror image of the other. Some figures have 2 or more lines of symmetry.

a. This lobster figure has a horizontal line of symmetry.b. This saucer figure has a vertical and horizontal line of symmetryc. This ying/yang figure has 2 congruent parts, but it has no line of symmetry

a. b. c.

Although this ying/yang figure lacks line symmetry, it does have Rotational Symmetry.To be rotational a figure must coincide with itself after rotating 180⁰ or less, either clockwise or counter-clockwise.

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Reflective Symmetry Rotational Symmetry

Must coincide with itself after rotating 180 ⁰or less

Mirror image of the other side of line of symmetry

1.4Exploring Symmetry

Line of Symmetry

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The midpoint of the line segment joining

A (x1, y1) and B (x2, y2) is as follows:

•A ( ─2, 5)

C ( 1, 3)

B ( 4, 1)

M = (x1 + x2

,y1 + y2

)2 2

Each coordinate of M is the mean of the corresponding coordinates of A and B.

M = (─2 + 4

,5 + 1

)2 2

M = (2

,6

)2 2

The Midpoint Formula 1.4

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To understand GEOMETRY you must first learn to speak the language

by studying the terminology

Then you need to practice applying learned principles in new problem situationsFinally you need to remember learned concepts as building blocks for new ones to follow

Next you need to understand the logic of geometry through

deductive and inductive reasoning

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The Building Blocks of Geometry

UNDEFINED TERMS : can’t be defined by simpler terms. [ Point , Line, Plane ]

DEFINED TERMS : can be defined by the undefined terms or previously defined terms so it is easier to

describe geometric figures and relationships

POSTULATES : a statement that is accepted without proof.

THEOREMS : a statement that is proven to be true.

The progression of undefined, defined terms, postulates and theorems lead to evolving new logical generalizations.

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y axis

Quadrant I

( + , + )

Quadrant II

( – , + )

Quadrant III

( – , – )Quadrant IV

( + , – )

x axis

( 0 , 0 )

Coordinate Plane

Coordinate & Noncoordinate Geometry

Ordered pairs in form of ( x , y ) x coordinate is firsty coordinate is second

1.5

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Slope and Rate of ChangeSLOPE of a non-vertical line is the ratio of a vertical change (RISE) to a horizontal change (RUN).

Slope of a line:m = y2 – y1 = RISE

x2 – x1 = RUN

(differences in y values)(differences in x values) •

y2 – y1

RISE

x2 – x1

RUNx

y

( x1 , y1 )

( x2 , y2 )•

1.5

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Slope and Rate of ChangeCLASSIFICATION OF LINES BY SLOPE

A line with a + slope rises from left to right [ m > 0 ]

A line with a – slope falls from left to right [ m < 0 ]

A line with a slope of 0 is horizontal [ m = 0 ]

A line with an undefined slope is vertical [ m = undefined, no slope ]

Positive Slope

Negative Slope

0 Slope

No Slope

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Slope and Rate of ChangeSLOPES OF PARALLEL & PERPENDICULAR LINES

PARALLEL LINES: the lines are parallel if and only if they have the SAME SLOPE.

m1 = m2

PERPENDICULAR LINES: the lines are perpendicular if and only if their SLOPES are NEGATIVE RECIPROCALS.

m1 = - 1 m2

m1 m2 = - 1

or

1.5

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Slope and Rate of Change

Ex 1: Find the slope of a line passing through ( – 3, 5 ) and ( 2, 1 )

Let ( x1, y1 ) = ( – 3, 5 ) and ( x2, y2 ) = ( 2, 1 )

y

x

( 2, 1 )

( - 3, 5 )•

5

– 4

Slope of a line:m = y2 – y1 = RISE (differences in y values)

x2 – x1 = RUN (differences in x values)

m = 5 – 1 = 4 – 3 – 2 – 5

m = 1 – 5 = – 4 2 + 3 5

OR

1.5

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Coordinate & Noncoordinate Geometry

Coordinate Geometry (also known as Analytic Geometry)

Shows a graph

Non-Coordinate Geometry (also known as Euclidean Geometry)

has no graph

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Perimeter & Area

Square RectangleP = 4 s and A = s2 P = 2 l + 2 w and A = l w

Triangle CircleP = a + b + c and A = ½ b h C = 2 ∏ r and A = ∏ r2

S

a

w

l

h

b

c

S

r

1.6

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Find the Perimeter & Area 1.6

Example 1

Each square on the grid at the left is 1 foot by 1 foot. Find the perimeter and area of the green region.

SOLUTION: The distance around the shaded figure is 40 units. Thus the perimeter of the region is 40 feet. By counting the shaded squares, you can find that the area is 36 square feet.

Can you think of another way to find the area?

Subtract the number of non-green units within the 6 x 8 = 48 total units.

Or, 48 – 12 = 36

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6 ft

6 ft

24 ft

40 ftArea of

sidewalk Area of larger

rectangle

Area of pool= ─

Finding an Area

= ( 36 ) ( 52 ) – ( 24 ) ( 40 )= 1872 – 960 = 912 square feet

What is the area of the sidewalk?

1.6

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Undefined Terms

Description Notation

Point A point indicates position; it has no length, width, or depth

• AA point is named by a single capital letter

Line A line is a set of continuous points that extend indefinitely in either direction

A BA line is identified by naming two points on the line and drawing a line over the letters.

↔ AB

PlaneA plane is a set of points that forms a flat surface that has no depth and that extends indefinitely in all directions

A plane is usually represented as a closed four-sided figure and is named by placing a capital letter at one of the corners.

Undefined Terms

• •

P

2.1

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B

P

L

kA

B●

This illustration shows that lines may lie in different planes or in the same plane.

Here lines L and AB are on plane “B” while lines k and AB are on plane “P”. NOTE: line AB lies in both “B and P” planes, which is also a intersecting line of these planes.

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Defined Terms

Description Illustration

Line segment

A line segment is a part of a line consisting of two points, called end points, and the set of all points between them. Notation: A B

RayA Ray is a part of a line consisting of a given point, called the end point and the set of all points on the one side of the end point.

A Ray is always named by using two points, the first of which must be the end point. The arrow above always points to the right.

Notation: LM

LineOr

Opposite Rays

Opposite rays have the same end point and form a line.

The line is indicated by KB or BKKX and KB are opposite rays.

Defined Terms

• •

B

A B

●L M

KX● ● ●

2.1

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Defined Terms

Description Illustration

Angle

An angle is the union of two rays having the same end point. The end point is called the vertex of the angle, and the rays are called the sides of the angle.

The measure of < A is denoted by m < A < 90. Angles are classified as acute, right, obtuse and straight.

Acute 0 ⁰ < m < A < 90 ⁰ 2 angles are adjacent if they share a common vertex and side, but have no common interior pointsRight m < A = 90 ⁰

Obtuse 90 ⁰ < m < A < 180 ⁰

Straight m < A < 180 ⁰

Reflex 180 ⁰ < m < A < 360 ⁰

Defined Terms

K

Vertex: K

Sides KJ and KL

●J

L●●

12

3

●V

< 1 and < 2 are adjacent< 1 and < 3 are not adjacent

2.1

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An angle may be named in one of three ways:

1. Using three letters, the center letter corresponding to the vertex of the angle and the other letters representing points on the sides of the angle. For example, in Figure 1, the name of the angle whose vertex is T can be angle RTB ( < RBT )

2. Placing a number of the vertex and in the interior of the angle. The angle may then be referred to by the number. For example, in Figure 2, the name of the angle whose vertex is T can be < 1 or < RTB or < BTR.

Figure 1 Figure 2

.

Naming Angles

B

R

T ●

● ●

R

T

B

Interior of angleExterio

r of angle

Exterior of angle

1)

2.1

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An angle may be named in one of three ways:

3. Using a single letter that corresponds to the vertex, provided that this does not cause any confusion. There is no question which angle on the diagram corresponds to angle A in figure 3, but which angle on the diagram is angle D?

Actually 3 angles are formed at vertex D:

1. angle ADB ( < ADB )2. angle CDB ( < CDB )

3. angle ADC ( < ADC)

.

Naming Angles

C

A D

B

)Figure 3

) )

In order to uniquely identify the angle having D as its vertex, we must either name the angle using three letters or introduce a number into the diagram.

2.1

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Example of Naming Angles

Use three letters to name each of the numbered angles in the accompanying diagram.

A

B C

D

E

12

3

4< 1 =< 2 = < 3 = < 4 =

M

L

2.1

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Example of Naming Lines and Line Segments

● ● ●J W R

Name the line in three different ways =

Name three different segments =

Name four different rays =

Name a pair of opposite rays =

2.1

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Definitions

The purpose of a definition is to make the meaning of a term clear. A good definition must:• Clearly identify the word or expression that is being defined• State the distinguishing characteristics of the term being defined, using

commonly understood or previously used• Be expressed in a grammatically correct sentence

Example: consider the terms collinear and non-collinear points

• Collinear points are points that lie on the same line• Non-collinear points are points that do not lie on the same line

●●

● ●●

●AC

B

S

T

R

Much of geometry involves building on previously discussed ideas. Example: we can use current knowledge of geometric terms to arrive at a definition of a triangle. How would you draw a triangle? If you start with 3 non-collinear points and connect them with line segments, a triangle is formed.

2.1

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Definitions

A good definition must be reversible as shown in the following table

●●●

●●

●The first two definitions are reversible since the reverse of the definition in a true statement. The reverse of the third definition is false since the points may be scattered as in example to right.

Definition Reverse of the Definition

Collinear points are points that lie in the same line.

Points that lie on the same line are collinear points.

A right angle is an angle whose measure is 90 ◦

An angle whose measure is 90 ◦ is a right angle.

A line segment is a set of points. A set of points is a line segment.

2.1