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

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.

The Elements

• Knowing that you can’t define everything and that you can’t prove everything, Euclid began by stating three undefined terms:  Point is that which has no part  (Straight) Line is a line that lies evenly with the points on itself  Plane (Surface) is a plane that lies evenly with the straight lines on itself Actually, Euclid did attempt to define these basic terms . . .

Basic Terms & Definitions

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

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

AB A B

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

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

• A right angle has a measure of 90 degrees.

• A straight angle has a measure of 180 degrees.

• 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.

# of sides 3 4 5 6 7 8 9 10

Polygons

name of Polygon tri angle quad rilateral penta gon hexa gon hepta gon octa gon nona gon deca gon

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.

kite

From General to Specific

Quadrilateral trapezoid parallelogram rhombus rectangle square

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.

area perimeter

a c

Triangle

b h Perimeter = a + b + c 1 Area = bh 2 The height of a triangle is measured perpendicular to the base.

Rectangle and Square

s w l Perimeter = 2w + 2l Area = lw Perimeter = 4s Area = s 2

Parallelogram

a h b Perimeter = 2a + 2b Area = hb  Area of a parallelogram = area of rectangle with width = h and length = b

c h a

Trapezoid

b d b a Perimeter = a + b + c + d Area = 1 2 h(a + b)  Parallelogram with base (a + b) and height = h with area = h(a + b)  But the trapezoid is half the parallelgram 

6 5

Ex: Name the polygon

2 1 3  hexagon 4 1 2 5  pentagon 3 4

Ex: What is the perimeter of a triangle with sides of lengths 1.5 cm, 3.4 cm, and 2.7 cm?

1.5

3.4

2.7

Perimeter = a + b + c = 1.5 + 2.7 + 3.4

= 7.6

Ex: The perimeter of a regular pentagon is 35 inches. What is the length of each side?

s Recall: a regular polygon is one with congruent sides.

Perimeter = 5s 35 = 5s s = 7 inches

Ex: A parallelogram has a based of length 3.4 cm. The height measures 5.2 cm. What is the area of the parallelogram? 5.2

3.4

Area = (base)(height) Area = (3.4)(5.2) = 17.86 cm 2

Ex: The width of a rectangle is 12 ft. If the area is 312 ft

2

, what is the length of the rectangle?

12 312 L Area = (Length)(width) Let L = Length 312 = (L)(12) L = 26 ft Check: Area = (Length)(width) = (12)(26) = 312

r

Circle

d • 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 = 2 p r • The area of a circle is A = p r 2 .

Find the Circumference

• The circumference, C, of a circle is the distance around the circle. C = 2 p r 1.5 cm • C = 2 p r • C = 2 p (1.5) • C = 3 p cm

Find the Area of the Circle

• The area of a circle is A = p r 2 8 in • d=2r • 8 = 2r • 4 = r • A = p r 2 • A = p(4) 2 • A = 16 p sq. in.

Composite Geometric Figures

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

• Ex:

+

• Ex: Composite Figure

-

8 Ex: Find the perimeter of the following composite figure 15

= +

Rectangle with width = 8 and length = 15 Half a circle with diameter = 8  radius = 4 Perimeter of partial rectangle Circumference of half a circle = (1/2)(2 p 4) = 4 p .

= 15 + 8 + 15 = 38 Perimeter of composite figure = 38 + 4 p .

Ex: Find the perimeter of the following composite figure 60 12 28 42 ? = b 12 28 ? = a 60 b a 42 60 = a + 42  a = 18 28 = b + 12  b = 16 Perimeter = 28 + 60 + 12 + 42 + b + a = 28 + 60 + 12 + 42 + 16 + 18 = 176

Ex: Find the area of the figure

3 3 3 8 Area of triangle = ½ (8)(3) = 12 8 3 8 Area of rectangle = (8)(3) = 24 Area of figure = area of the triangle + area of the square = 12 + 24 = 36.

Ex: Find the area of the figure

4 4 3.5

3.5

4 Area of rectangle = (4)(3.5) = 14 The area of the figure = area of rectangle – cut out area = 14 – 2 p square units.

Diameter = 4  radius = 2 Area of circle = p 2 2 = 4 p  Area of half the circle = ½ (4 p ) = 2 p

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?

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

30 Width = 2 + 20 + 2 = 24 20 2 Length = 2 + 30 + 2 = 34 Therefore, the big rectangle has area = (24)(34) = 816 m 2 .

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

The small rectangle has area = (20)(30) = 600 m 2 .

20 by 30 The area of the walkway is the difference between the big and small rectangles: Area = 816 – 600 = 216 m 2 .

Find the area of the shaded region 10 10 Area of square = 10 2 = 100 Area of each circle = p 5 2 = 25 p 10 r = 5 ¼ of the circle cuts into the square. r = 5 But we have four ¼ 4(¼)(25 p ) cuts into the area of the square.

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