Chapter 12 Geometric Shapes Section 12.1 Recognizing Shapes The van Hiele Theory Level 0 (Recognition) At this lowest level, a child recognizes certain

Chapter 12 Geometric Shapes

Section 12.1 Recognizing Shapes

The van Hiele Theory

Level 0 (Recognition)At this lowest level, a child recognizes certain shapes holistically without paying attention to their components

Level 1 (Analysis)The child focuses analytically on the parts of a figure, such as its sides and angles. Component parts and their attributes are used to describe and characterize figures. Relevant attributes are understood and are differentiated from irrelevant attributes.

Level 2 (Relationships)There are two types of thinking at this level. First, a child understands abstract relationships among figures. For example, a square is both a rhombus and a rectangle. Second, a child can use informal deductions to justify observations made at level 1. For instance, a rhombus is also a parallelogram.

Section 12.1 Recognizing Shapes

The van Hiele Theory

Level 3 (Deduction)Reasoning at this level includes the study of geometry as a formal mathematical system. A student at this level can understand the notions of mathematical postulates and theorems.

Level 4 (Axiomatics)Geometry at this level is highly abstract and does not necessarily involve concrete or pictorial models. The postulates or axioms themselves become the object of intense, rigorous scrutiny. This level of study is only suitable for university students.

Describing Common Geometric Shapes

Model Abstraction Description

Top of a window Line segment

Open pair of scissors Angle The union of two line segments with a common endpoint.

Vertical flag pole Right angle Angle formed by two lines, one vertical and one horizontal.



Bike frame Scalene triangle Triangle with all 3 sides different.

Teepee Isosceles triangle Triangle with 2 sides the same length.

Yield sign Equilateral triangle Triangle with all 3 sides the same length.



Steel frame of a bridge

Right triangle Triangle with one right angle.

Tile Square Quadrilateral with all sides the same length and 4 right angles.

Door Rectangle Quadrilateral with 4 right angles.

Picture on next slide

Steel bridge in Portland



Railing Parallelogram Quadrilateral with 2 pairs of parallel sides

Diamond Rhombus Quadrilateral with 4 sides the same length.

Kite Kite Quadrilateral with two non-overlapping pairs of adjacent sides that are of the same length



Trapezoid Quadrilateral with exactly one pair of parallel sides.

Paper cup(silhouette)

Isosceles Trapezoid Trapezoid whose non-parallel sides are of the same length.

Summary

Quadrilateral

Trapezoid

Isosceles Trapezoid

Parallelogram Kite

Rectangle Rhombus

Square

Section 12.2 Analyzing Shapes

Category 1 Category 2

What is the mathematical property that separates these two categories of shapes?

Symmetries

In formal terms, we say that an object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance.

Reflection Symmetry (also called folding symmetry)

A 2D figure has reflection symmetry if there is a line that the figure can be “folded over” so that one-half of the figure matches the other half perfectly.

The “fold line” just described is call the figure’s line (axis) of symmetry.

Lines of symmetry for the following common figures.

Rotation Symmetry

A 2D figure has rotation symmetry if there is a point around which the figure can be rotated, less than a full turn, so that the image matches the original figure perfectly.

(click to see animation)

This equilateral triangle has 2 (non-trivial) rotation symmetries, 120° and 240° respectively. Since every figure will match itself after rotating 360°, we do not consider a 360° rotation as a rotation symmetry.

Rotation symmetries of common figures

Rectangle(1 symmetry)

Square(3 symmetries) Diamond

(1 symmetry)

Parallelogram(1 symmetry)

Regular Pentagon(4 symmetries)

Trapezoid(no symmetry)

We don’t count the trivial 360° rotation symmetry here.

Polygons

The word "polygon" derives from the Greek poly, meaning "many", and gonia, meaning "angle".

Equilateral triangle

n = 3

Square n = 4

Regularpentagon

n = 5

Regularhexagon

n = 6

Regularheptagon

n = 7

Regularoctagon

n = 8

Polygons and their nomenclature

A Triangle (from Latin) has 3 sides

A Quadrilateral (from Latin) has 4 sides

A Pentagon (from Greek) has 5 sides

A Hexagon (from Latin) has 6 sides

A Heptagon (from Greek) (or a Septagon from Latin?) has 7 sides

In fact, “Septagon” is not an official word for the 7-gon, it is not even in a dictionary. It was invented by some elementary school teachers to make it easier to remember. The Latin word septem means 7 and September means the seventh month.The old Roman calendar began the year in January, (named after the Roman god of fortune, Janus), and September was the seventh month. Afterwards, Julius Augustus (46 BC) named two more then-29 day periods after himself and September came to be as we know it in the Gregorian Calendar, the ninth month.

An Octagon (from Greek) has 8 sides

A Nonagon (from Latin) has 9 sides.

A Decagon (from Greek) has 10 sides.

More names of polygons

A polygon with more than n (>10) sides is usually just called an n-gon.

Convex and Concave Shapes

A figure is convex if a line segment joining any two points inside the figure lies completely inside the figure.

Angles in a polygon

In a regular pentagon:

the measure of a central angle is 360°/5 = 72°the measure of an exterior angle is also 360°/5 = 72°the measure of a vertex angle is 180° – 72° = 108°

CirclesA Circle is the set of all points in the plane that are at a fixed distance from a given point called the center.

The distance from any point on the circle to the center is called the radius of the circle.

The length of any line segment whose endpoints are on the circle and which contains the center is called the diameter of the circle. The segment is also called a diameter of the circle.

CirclesCircles have the following 3 properties that make them very useful.

1. They are highly symmetrical, hence they have a sense of beauty and are often used in designs.

eg. dinnerware.

CirclesCircles have the following 3 properties that make them very useful.

1. They are high symmetrical, hence they have a sense of beauty and are often used in designs.

eg. dinnerware.

2. Every point on a circle bears the same distance from the center. This is called the equidistance property.

Applications: wheels

3. For a given (fixed) perimeter, the circle has the largest area.

Applications: soda cans, or any container for pressurized liquid are all cylindrical in shape.

Section 12.3 Properties of Lines and Angles

DefinitionTwo different given lines L1 and L2 on a plane are said to be parallel if they will never intersect each other no matter how far they are extended.

Angles

An angle is the union of two rays with a common endpoint.

vertex

side

side

interior

Degrees

One degree is divided into 60 minutes and one minute is further divided into 60 seconds.

Notations:For instance, 27 degrees 35 minutes 41 seconds is written as 27°35’41”

Angles are measured by a semi-circular device called a protractor.

The whole circle is divided into 360 equal parts, each part is defined to have measure one degree (written 1°). Hence a semi-circular protractor has 180 degrees.

Names of angles

A straight angle has 180 degrees

An obtuse angle has measure between 90° and 180°.

A right angle has exactly 90°.

An acute angle has measure less than 90°.

DefinitionTwo angles are called vertical angles if they are opposite to each otherand are formed by a pair of intersecting lines.

A B

TheoremAny pair of vertical angles are always congruent.

More special angles

Two angles are said to be supplementary if their measures add up to 180°.

Two angles are said to be complementary if their measures add up to 90°.

α β

αβ

Perpendicular Lines

Two lines are said to be perpendicular to each other if they intersect to form a right angle

Parallel Lines and Angles

DefinitionGiven two line L1 and L2 (not necessarily parallel) on the plane, a third line T is called a transversal of L1 and L2 if it intersects these two lines.

L1

L2

T

L1

L2

T

DefinitionsLet L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal.a) a and form a pair of corresponding angles.b) c and form a pair of corresponding angles etc.

a

c

L1

L2

T

DefinitionsLet L1 and L2 be two lines (not necessarily parallel) on the plane, and T

be a transversal.c) c and form a pair of alternate interior angles.d) d and form a pair of alternate interior angles.

cd

L1

L2

T

DefinitionsLet L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal.e) a and form a pair of alternate exterior angles.f) b and form a pair of alternate exterior angles.

a

Angle Sum in a Triangle

a

b

c

Draw an arbitrary triangle on a piece of paper and label all 3 angles. Next cut out the triangle, and then cut it into 3 parts (as indicated by the dashed lines)

Arrange the 3 angles side by side, can you get a straight angle?

Conclusion:The angle sum in a triangle is always 180°

Angle Sum in other Polygons

What is the sum of all angles in a quadrilateral?

Answer: 180 × 2 = 360

What is the sum of all angles in a pentagon?

Answer: 180 × 3 = 540

Angle Sum in other Polygons

Conclusion:For a polygon with n sides, the angle sum is

(n – 2) × 180°

Classification of triangles according to their angles.

A triangle with one right angle is called a right triangle.

A triangle with one obtuse angle is called an obtuse triangle.

A triangle with 3 acute angles is called an acute triangle.

Classification of triangles according to their sides.

A triangle with 3 equal sides is called

an equilateral triangle.

A triangle with 2 equal sides is called an isosceles triangle.

A triangle with 3 different sides is called a

scalene triangle.

Angles & Angle Sums in Regular polygons

center

vertex angle

cent

ral a

ngle

For a regular pentagon,

m(central angle) =

725

360

m(vertex angle) = (3 × 180) ÷ 5 = 108

Application of Degree MeasureAngles can be used to indicate directions. The only difference is that the measure can be greater than 180º.

In navigation, the direction can be any value between 0º and 360º.

The Bearing SystemThe exact (magnetic) North is defined to be 0 degree.

Any other direction is defined to be the number of degrees away from exact North measuring in the

clockwise direction.

N

south east direction

130º

The Bearing SystemIn particular, 90º is equal to exact East,

N

90º = East

The Bearing Systemand 180º is equal to exact South,

N

180º = South

The Bearing Systemand 270º is equal to exact West,

N

270º = West

Runway NumbersIn any airport, each runway is assigned a number according to the direction it is pointing at – except that the units digit is omitted for simplicity.

For example, runway 24 is actually pointing at 240º, and it means that during final approach, the aircraft is heading 240º - which is about south west.

3

21

This is one of the many signs that you will see in a big commercial airport. It tells the pilots which runway is in front of them.

Exercise:Fill in the missingrunway numbers.






Airports around San Diego

San Diego International

Montgomery Field

Gillespe Field

San Diego International

Montgomery Field

Section 12.4 Regular Polygons and Tessellations

Tessellations (or Tilings)A tessellation is an arrangement of congruent shapes that cover an entire area with no overlaps or gaps.

A 2D geometric figure R is said to tessellate (or tile) the plane if the entire plane can be completely covered by (an infinite number of) congruent copies of R with no overlaps or gaps.

We can also tile a plane with congruent copies of several different polygons. These are called semiregular tessellations

Convex and Concave Polygons

A polygon X is said to be convex if you take any two points on X (including the boundary), the line segment joining them lies entirely within the tile (again including the boundary).

a convex quadrilateral a concave (i.e. non-convex) quadrilateral

Question: What polygons can tessellate the plane?

1. Any triangle can tessellate the plane.2. Any square can tessellate the plane.3. Any rectangle can tessellate the plane.4. Any convex quadrilateral can tessellate the plane.5. In fact, any quadrilateral (including non-convex ones) can tessellate

the plane.6. A regular pentagon will not tessellate the plane.7. Any regular hexagon can tessellate the plane.8. In fact, exactly 3 classes of convex hexagons can tile the plane.

(this was proved by K. Reinhardt in his 1918 doctoral thesis. He also went on to explore the tessellations by irregular but convex pentagons and found 5 classes that do tile the plane.He felt that he had found all of them even though he could not give a proof because he claimed that it would be very tedious to do so.)

In 1968, after 35 years working on the problem on and off, R. B. Kershner, a physicist at Johns Hopkins University, discovered 3 more classes of pentagons that will tessellate.Kershner was sure that he had found all of them, but again did not offer a complete proof, which “would require a rather big book.”

Shortly after an article of the “complete” classification of convex pentagons into 8 types appeared in Scientific American (July 1975), an amateur mathematician (R. James III) discovered a 9th type!Between 1976 and 1977, a San Diego housewife Marjorie Rice, without formal education in mathematics beyond high school, found 4 more types!

A 14th type was found by a mathematics graduate student in 1985. Since then, no new types have been found, and yet no one knows if the classification is complete.

(the 14 types of pentagons that tile the plane)

With the situation so intricate for convex pentagons, you might think that it must be even worse for polygons with 7 or more sides. However, the situation is remarkably simple, as Reinhardt proved in 1927:

A convex polygon with 7 or more sides cannot tessellate.

Section 12.5 Describing 3-Dimensional Shapes

(This will be taught after section 13.2)

Documents

Chapter 12 Geometric Shapes Section 12.1 Recognizing Shapes The van Hiele Theory Level 0 (Recognition) At this lowest level, a child recognizes certain