introduction to geometry

32
INTRODUCTION TO GEOMETRY MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

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INTRODUCTION TO GEOMETRY. MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur. Geometry. The word geometry comes from Greek words meaning “to measure the Earth” Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics. - PowerPoint PPT Presentation

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Page 1: INTRODUCTION TO GEOMETRY

INTRODUCTION TO GEOMETRY

MSJC ~ San Jacinto CampusMath Center Workshop Series

Janice Levasseur

Page 2: INTRODUCTION TO GEOMETRY

Geometry

• The word geometry comes from Greek words meaning “to measure the Earth”

• Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics

Page 3: INTRODUCTION TO GEOMETRY

The Greeks and Euclid

• Our modern understanding of geometry began with the Greeks over 2000 years ago.

• The Greeks felt the need to go beyond merely knowing certain facts to being able to prove why they were true.

• Around 350 B.C., Euclid of Alexandria wrote The Elements, in which he recorded systematically all that was known about Geometry at that time.

Page 4: INTRODUCTION TO GEOMETRY

Basic Terms & Definitions

• A ray starts at a point (called the endpoint) and extends indefinitely in one direction.

• A line segment is part of a line and has two endpoints.

A B AB

BA AB

Page 5: INTRODUCTION TO GEOMETRY

• An angle is formed by two rays with the same endpoint.

• An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees.

vertex

side

side

Page 6: INTRODUCTION TO GEOMETRY

• A right angle has a measure of 90 degrees.

• A straight angle has a measure of 180 degrees.

Page 7: INTRODUCTION TO GEOMETRY

• A simple closed curve is a curve that we can trace without going over any point more than once while beginning and ending at the same point.

• A polygon is a simple closed curve composed of at least three line segments, called sides. The point at which two sides meet is called a vertex.

• A regular polygon is a polygon with sides of equal length.

Page 8: INTRODUCTION TO GEOMETRY

Polygons

# of sides name of Polygon3 triangle

4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon

Page 9: INTRODUCTION TO GEOMETRY

Quadrilaterals• Recall: a quadrilateral is a 4-sided polygon. We can

further classify quadrilaterals: A trapezoid is a quadrilateral with at least one pair of

parallel sides. A parallelogram is a quadrilateral in which both pairs of

opposite sides are parallel. A kite is a quadrilateral in which two pairs of adjacent

sides are congruent. A rhombus is a quadrilateral in which all sides are

congruent. A rectangle is a quadrilateral in which all angles are

congruent (90 degrees) A square is a quadrilateral in which all four sides are

congruent and all four angles are congruent.

Page 10: INTRODUCTION TO GEOMETRY

From General to Specific

Quadrilateral

trapezoidkite

parallelogram

rhombus

rectangle

square

Mo

re s

pe

cific

Page 11: INTRODUCTION TO GEOMETRY

Perimeter and Area

• The perimeter of a plane geometric figure is a measure of the distance around the figure.

• The area of a plane geometric figure is the amount of surface in a region.

perimeter

area

Page 12: INTRODUCTION TO GEOMETRY

Triangleh

b

a c

Perimeter = a + b + c

Area = bh21

The height of a triangle is measured perpendicular to the base.

Page 13: INTRODUCTION TO GEOMETRY

Rectangle and Square

w

l

s

Perimeter = 2w + 2l Perimeter = 4s

Area = lw Area = s2

Page 14: INTRODUCTION TO GEOMETRY

Parallelogram

b

a h

Perimeter = 2a + 2b

Area = hb Area of a parallelogram = area of rectangle with width = h and length = b

Page 15: INTRODUCTION TO GEOMETRY

Trapezoid

c d

a

bPerimeter = a + b + c + d

Area =

b

a

Parallelogram with base (a + b) and height = h with area = h(a + b) But the trapezoid is half the parallelgram

h(a + b)21

h

Page 16: INTRODUCTION TO GEOMETRY

Ex: Name the polygon

3

21

45

6

hexagon

1

2

34

5 pentagon

Page 17: INTRODUCTION TO GEOMETRY

Ex: What is the perimeter of a triangle with sides of lengths 1.5

cm, 3.4 cm, and 2.7 cm?

1.5 2.7

3.4

Perimeter = a + b + c

= 1.5 + 2.7 + 3.4

= 7.6

Page 18: INTRODUCTION TO GEOMETRY

Ex: The perimeter of a regular pentagon is 35 inches. What is the

length of each side?

Perimeter = 5s

35 = 5s

s = 7 inches

s

Recall: a regular polygon is one with congruent sides.

Page 19: INTRODUCTION TO GEOMETRY

Ex: A parallelogram has a based of length 3.4 cm. The height

measures 5.2 cm. What is the area of the parallelogram?

3.4

5.2

Area = (base)(height)

Area = (3.4)(5.2)

= 17.86 cm2

Page 20: INTRODUCTION TO GEOMETRY

Ex: The width of a rectangle is 12 ft. If the area is 312 ft2, what is the length of the rectangle?

12 312

Area = (Length)(width)

L = 26 ft

Let L = Length

L312 = (L)(12)

Check: Area = (Length)(width) = (12)(26)

= 312

Page 21: INTRODUCTION TO GEOMETRY

Circle

• A circle is a plane figure in which all points are equidistance from the center.

• The radius, r, is a line segment from the center of the circle to any point on the circle.

• The diameter, d, is the line segment across the circle through the center. d = 2r

• The circumference, C, of a circle is the distance around the circle. C = 2r

• The area of a circle is A = r2.

r

d

Page 22: INTRODUCTION TO GEOMETRY

Find the Circumference

• The circumference, C, of a circle is the distance around the circle. C = 2r

• C = 2r• C = 2(1.5)• C = 3cm

1.5 cm

Page 23: INTRODUCTION TO GEOMETRY

Find the Area of the Circle• The area of a circle is A = r2

• d=2r• 8 = 2r• 4 = r

• A = r2

• A = 2

• A = 16sq. in.

8 in

Page 24: INTRODUCTION TO GEOMETRY

Composite Geometric Figures

• Composite Geometric Figures are made from two or more geometric figures.

• Ex:

+

Page 25: INTRODUCTION TO GEOMETRY

• Ex: Composite Figure

-

Page 26: INTRODUCTION TO GEOMETRY

Ex: Find the perimeter of the following composite figure

+=

8

15

Rectangle with width = 8 and length = 15

Half a circle with diameter = 8 radius = 4

Perimeter of composite figure = 38 + 4.

Perimeter of partial rectangle = 15 + 8 + 15 = 38

Circumference of half a circle = (1/2)(24) = 4.

Page 27: INTRODUCTION TO GEOMETRY

Ex: Find the perimeter of the following composite figure

28

60

42

12

? = a

? = b

60

a 42

60 = a + 42 a = 18

28

b

12

28 = b + 12 b = 16

Perimeter = 28 + 60 + 12 + 42 + b + a = 28 + 60 + 12 + 42 + 16 + 18 = 176

Page 28: INTRODUCTION TO GEOMETRY

Ex: Find the area of the figure

3

3

8

8

Area of rectangle = (8)(3) = 24

3

8

Area of triangle = ½ (8)(3) = 12

Area of figure = area of the triangle + area of the square = 12 + 24 = 36.

3

Page 29: INTRODUCTION TO GEOMETRY

Ex: Find the area of the figure4

3.5

4

3.5

Area of rectangle = (4)(3.5) = 14

4

Diameter = 4 radius = 2

Area of circle = 22 = 4 Area of half the circle = ½ (4) = 2

The area of the figure = area of rectangle – cut out area

= 14 – 2 square units.

Page 30: INTRODUCTION TO GEOMETRY

Ex: A walkway 2 m wide surrounds a rectangular plot of grass.

The plot is 30 m long and 20 m wide. What is the area of the walkway?

20

302

What are the dimensions of the big rectangle (grass and walkway)?

Width = 2 + 20 + 2 = 24

Length = 2 + 30 + 2 = 34

The small rectangle has area = (20)(30) = 600 m2.

What are the dimensions of the small rectangle (grass)?

Therefore, the big rectangle has area = (24)(34) = 816 m2.

The area of the walkway is the difference between the big and small rectangles:

20 by 30

Area = 816 – 600 = 216 m2.

2

Page 31: INTRODUCTION TO GEOMETRY

Find the area of the shaded region10

10

10 r = 5

Area of each circle = 52 = 25

¼ of the circle cuts into the square.

But we have four ¼

4(¼)(25) cuts into the area of the square.

Area of square = 102 = 100

Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25 square units

r = 5

Page 32: INTRODUCTION TO GEOMETRY